JaCoP Library
User’s Guide

__________________________________________________________________

Krzysztof Kuchcinski and Radosław Szymanek

Version 4.4,  April 11, 2016

Copyright ©2002–2016 Krzysztof Kuchcinski and Radosław Szymanek. All rights reserved.

Contents

1 Introduction
 1.1 Getting started
2 Using JaCoP library
 2.1 Finite Domain Variables
 2.2 Finite domains
 2.3 Constraints
 2.4 Search for solutions
3 Constraints
 3.1 Primitive constraints
 3.2 Logical and conditional constraints
 3.3 Global Constraints
  3.3.1 Alldifferent constraints
  3.3.2 Circuit constraint
  3.3.3 Subcircuit constraint
  3.3.4 Element constraint
  3.3.5 Distance constraint
  3.3.6 Cumulative constraint
  3.3.7 Diff2 constraint
  3.3.8 Min, Max and ArgMin, ArgMax constraints
  3.3.9 SumInt, SumBool and LinearInt constraints
  3.3.10 ExtensionalSupport and ExtensionalConflict constraints
  3.3.11 Assignment constraint
  3.3.12 Count constraint
  3.3.13 Values constraint
  3.3.14 Global cardinality constraint (GCC)
  3.3.15 Among and AmongVar
  3.3.16 Regular constraint
  3.3.17 Knapsack constraint
  3.3.18 Geost constraint
  3.3.19 NetworkFlow constraint
  3.3.20 Binpacking
  3.3.21 LexOrder
 3.4 Decomposed constraints
  3.4.1 Sequence constraint
  3.4.2 Stretch constraint
  3.4.3 Lex constraint
  3.4.4 Soft-Alldifferent
  3.4.5 Soft-GCC
4 Set Solver
 4.1 Set Variables and Set Domains
 4.2 Set Constraints
 4.3 Set Search
5 Floating Point Solver
 5.1 Floating Point Variables and Floating Point Domains
 5.2 Floating Point Constraints
 5.3 Floating Point Search
 5.4 Example
 5.5 Experimental Extensions
  5.5.1 Derivatives
  5.5.2 Multivariate Interval Newton Method
6 Search
 6.1 Depth First Search
  6.1.1 Restart search
 6.2 Search plug-ins
 6.3 Credit search
 6.4 Limited discrepancy search
 6.5 Combining search
7 SAT solver
 7.1 Boolean expressions translations
 7.2 FDV encoding and constraint translations
8 Flatzinc
 8.1 Minizinc Global Constraints
 8.2 Minizinc Compilation and Execution
 8.3 Flatzinc Extensions
  8.3.1 Flatzinc Loader
  8.3.2 Floating-Point Minimization
9 Scala Wrapper
 9.1 Example
 9.2 Scala Model
 9.3 Variables
 9.4 Constraints
 9.5 Search
 9.6 Other Methods
A JaCoP constraints
 A.1 Arithemtic constraints
 A.2 Set constraints
 A.3 Floating point constraints
 A.4 Logical, conditional and reified constraints
 A.5 Global constraints
B JaCoP search methods
 B.1 Variable and value selection for IntVar
 B.2 Variable and value selection for SetVar
 B.3 Variable selection for FloatVar
 B.4 Search methods
 B.5 Important methods for search plug-ins
C JaCoP debugging facilities
 C.1 Available switches
 C.2 CPviz interface

Chapter 1
Introduction

JaCoP library provides constraint programming paradigm implemented in Java. It provides primitives to define finite domain (FD) variables, constraints and search methods. The user should be familiar with constraint (logic) programming (CLP) to be able to use this library. A good introduction to CLP can be found, for example, in [15].

JaCoP library provides most commonly used primitive constraints, such as equality, inequality as well as logical, reified and conditional constraints. It contains also number of global constraints. These constraints can be found in most commercial CP systems [521619]. Finally, JaCoP defines also decomposable constraints, i.e., constraints that are defined using other constraints and possibly auxiliary variables.

JaCoP library can be used by providing it as a JAR file or by specifying access to a directory containing all JaCoP classes. An example how program Main.java, which uses JaCoP, can be compiled and executed in the Linux like environment is provided below.

   javac -classpath .:path_to_JaCoP Main.java  
   java -classpath .:path_to_JaCoP Main

or

   javac -classpath .:JaCoP.jar Main.java  
   java -classpath .:JaCoP.jar Main

Alternatively one can specify the class path variable.

In Java application which uses JaCoP it is required to specify import statements for all used classes from JaCoP library. An example of the import statements that import the whole subpackages of JaCoP at once is shown below.

   import org.jacop.core.*;  
   import org.jacop.constraints.*;  
   import org.jacop.search.*;

Obviously, different Java IDE (Eclipse, NetBeans, etc.) and pure Java build tools (e.g., Ant) can be used for JaCoP based application development.

1.1 Getting started

Consider the problem of graph coloring as depicted in Fig. 1.1. Below, we provide a simplistic program with hard-coded constraints and specification of the search method to solve this particular graph coloring problem.


PIC

Figure 1.1: An example graph.


import org.jacop.core.*; 
import org.jacop.constraints.*; 
import org.jacop.search.*; 
 
public class Main { 
 
    static Main m = new Main (); 
 
    public static void main (String[] args) { 
        Store store = new Store();  // define FD store 
        int size = 4; 
        // define finite domain variables 
        IntVar[] v = new IntVar[size]; 
        for (int i=0; i<size; i++) 
            v[i] = new IntVar(store, "v"+i, 1, size); 
        // define constraints 
        store.impose( new XneqY(v[0], v[1]) ); 
        store.impose( new XneqY(v[0], v[2]) ); 
        store.impose( new XneqY(v[1], v[2]) ); 
        store.impose( new XneqY(v[1], v[3]) ); 
        store.impose( new XneqY(v[2], v[3]) ); 
 
        // search for a solution and print results 
        Search<IntVar> search = new DepthFirstSearch<IntVar>(); 
        SelectChoicePoint<IntVar> select = 
            new InputOrderSelect<IntVar>(store, v, 
                                         new IndomainMin<IntVar>()); 
        boolean result = search.labeling(store, select); 
 
        if ( result ) 
            System.out.println("Solution: " + v[0]+", "+v[1] +", "+ 
                                              v[2] +", "+v[3]); 
        else 
            System.out.println("*** No"); 
    } 
}

This program produces the following output indicating that vertices v0, v1 and v3 get different colors (1, 2 and 3 respectively), while vertex v3 is assigned color number 1.

Solution: v0=1, v1=2, v2=3, v3=1

Chapter 2
Using JaCoP library

The problem is specified with the help of variables (FDVs) and constraints over these variables. JaCoP support both finite domain variables (integer variables) and set variables. Both variables and constraints are stored in the store (Store). The store has to be created before variables and constraints. Typically, it is created using the following statement.

   Store store = new Store();

The store has large number of responsibilities. In short, it knits together all components required to model and solve the problem using JaCoP (CP methodology). One often abused functionality is printing method. The store has redefined the method toString(), but use it with care as printing large stores can be a very slow/memory consuming process.

In the next sections we will describe how to define FDVs and constraints.

2.1 Finite Domain Variables

Variable X :: 1..100 is specified in JaCoP using the following general statement (assuming that we have defined store with name store). Clerly, we required to have created store before we can create variables as any constructor for variable will require providing the reference to store in which the variable is created.

   IntVar x = new IntVar(store, "X", 1,100);

One can access the actual domain of FDV using the method dom(). The minimal and maximal values in the domain can be accessed using min() and max() methods respectively. The domain can contain “holes”. This type of the domain can be obtained by adding intervals to FDV domain, as provided below:

   IntVar x = new IntVar(store, "X", 1, 100); 
   x.addDom(120, 160);

which represents a variable X with the domain 1..100 120..160.

FDVs can be defined without specifying their identifiers. In this case, a system will use an identifier that starts with “_” followed by a generated unique sequential number of this variable, for example “_123”. This is illustrated by the following code snippet.

   IntVar x = new IntVar(store, 1, 100);

FDVs can be printed using Java primitives since the method toString() is redefined for class Variable. The following code snippet will first create a variable with the domain containing values 1, 2, 14, 15, and 16.

   IntVar x = new IntVar(store, "x", 1, 2); 
   x.addDom(14, 16); 
   System.out.println(x);

The last line of code prints a variable, which produces the following output.

   X::{1..2, 14..16}

One special variable class is a BooleanVariable. They have been added to JaCoP as they can be handled more efficiently than FDVs with multiple elements in their domain. They can be used as any other variable. However, some constraints may require BooleanVariables as parameters. An example of Boolean variable definition is shown below.

   BooleanVar bv = new BooleanVar(s, "bv");

2.2 Finite domains

In the previous section, we have defined FDVs with domains without considering domain representation. JaCoP default domain (cllaed IntervalDomain) is represented as an ordered list of intervals. Each interval is represented by a pair of integers denoting the minimal and the maximal value. This representation makes it possible to define all possible finite domains of integers but it is not always computationally efficient. For some problems other representations might be more computationally efficient. Therefore, JaCoP also offers domain that is restricted to represent only one interval with its minimal and maximal value. This domain is called BoundDomain and can be used by a finite domain variable in a same way as interval domain. The only difference is that any attempt to remove values from inside the interval of this domain will have no effect.

The following code creates variable v with bound domain 1..10.

   IntVar v = new IntVar(s, "v", new BoundDomain(1, 10) );

2.3 Constraints

In JaCoP, there are three major types of constraints:

Primitive constraints and global constraints can be imposed using impose method, while decomposable constraints are imposed using imposeDecomposition method. An example that imposes a primitive constraint XneqY is defined below. Again, in order to impose a constraint a store object must be available.

   store.impose( new XeqY(x1, x2));

Alternatively, one can define first a constraint and then impose it, as shown below.

   PrimitiveConstraint c = new XeqY(x1, x2); 
   c.impose(store);

Both methods are equivalent.

The methods impose(constraint) and constraint.impose(store) often create additional data structures within the constraint store as well as constraint itself. Do note that constraint imposition does not involve checking if the constraint is consistent. Both methods of constraint imposition does not check whether the store remains consistent. If checking consistency is needed, the method imposeWithConsistency(constraint) should be used instead. This method throws FailException if the store is inconsistent. Note, that similar functionality can be achieved by calling the procedure store.consistency() explicitly (see section 6).

Constraints can have another constraints as their arguments. For example, reified constraints of the form X = Y B can be defined in JaCoP as follows.

   IntVar x = new IntVar(store, "X", 1, 100); 
   IntVar y = new IntVar(store, "Y", 1, 100); 
   IntVar b = new IntVar(store, "B", 0, 1); 
   store.impose( new Refified( new XeqY(x, y), b));

In a similar way disjunctive constraints can be imposed. For example, the disjunction of three constraints can be defined as follows.

   PrimitiveConstraint[] c = {c1, c2, c3}; 
   store.impose( new Or(c) );

or

   ArrayList<PrimitiveConstraint> c = 
                          new ArrayList<PrimitiveConstraint>(); 
   c.add(c1); c.add(c2); c.add(c3); 
   store.impose( new Or(c) );

Note, that disjunction and other similar constraints accept only primitive constraints as parameters.

2.4 Search for solutions

After specification of the model consisting of variables and constraints, a search for a solution can be started. JaCoP offers a number of methods for doing this. It makes it possible to search for a single solution or to try to find a solution which minimizes/maximizes given cost function. This is achieved by using the depth-first-search together with constraint consistency enforcement.

The consistency check of all imposed constrains is achieved calling the following method from class Store.

   boolean result = store.consistency();

When the procedure returns false then the store is in inconsistent state and no solution exists. The result true only indicates that inconsistency cannot be found. In other words, since the finite domain solver is not complete it does not automatically mean that the store is consistent.

To find a single solution the DepthFirstSearch method can be used. Since the search method is used both for finite domain variables and set variables it is recommended to specify the type of variables that are used in search. For finite domain variables, this type is usually . It is possible to not specify these information but it will generate compilation warnings if compilation option -Xlint:unchecked is used. A simple use of this method is shown below.

   IntVar[] vars; 
   ... 
   Search<IntVar> label = new DepthFirstSearch<IntVar>(); 
   SelectChoicePoint<IntVar> select = 
            new InputOrderSelect<IntVar>(store, 
                                   vars, new IndomainMin<IntVar>()); 
   boolean result = label.labeling(store, select);

The depth-first-search method requires the following information:

Different classes can be used to implement SelectChoicePoint interface. They are summarized in Appendix B. The following example uses SimpleSelect that selects variables using the size of their domains, i.e., variable with the smallest domain is selected first.

   IntVar[] vars; 
   ... 
   Search<IntVar> label = new DepthFirstSearch<IntVar>(); 
   SelectChoicePoint<IntVar> select = 
                new SimpleSelect<IntVar>(vars, 
                                         new SmallestDomain<IntVar>(), 
                                         new IndomainMin<IntVar>()); 
   boolean result = label.labeling(store, select);

In some situations it is better to group FDVs and assign the values to them one after the other. JaCoP supports this by another variable selection method called SimpleMatrixSelect. An example of its use is shown below. This choice point selection heuristic works on two-dimensional lists of FDVs.

   IntVar[][] varArray; 
   ... 
   Search<IntVar> label = new DepthFirstSearch<IntVar>(); 
   SelectChoicePoint<IntVar> select = 
                 new SimpleMatrixSelect<IntVar>( 
                                       varArray, 
                                       new SmallestMax<IntVar>(), 
                                       new IndomainMin<IntVar>()); 
   boolean result = label.labeling(store, select);

The optimization requires specification of a cost function. The cost function is defined by a FDV that, with the help of attached (imposed) constraints, gets correct cost value. A typical minimization for defined constraints and a cost FDV is specified below.

   IntVar[] vars; 
   IntVar cost; 
   ... 
   Search<IntVar> label = new DepthFirstSearch<IntVar>(); 
   SelectChoicePoint<IntVar> select = new SimpleSelect<IntVar>(vars, 
                                       new SmallestDomain<IntVar>(), 
                                       new IndomainMin<IntVar>()); 
   boolean result = label.labeling(store, select, cost);

JaCoP offers number of different search heuristics based on depth-first-search. For example, credit search and limited discrepancy search. They are implemented using plug-in listeners that modify the standard depth-first-serch. For more details, see section 6.

Chapter 3
Constraints

3.1 Primitive constraints

JaCoP offers a set of primitive constraints that include basic arithmetic operations (+,-,*,∕) as well as basic relations (=,,<,,>,). Subtraction and division are not provided explicitly, but since constraints define relations between variables, they are defined using addition and multiplication. For detailed list of primitive constraint see appendix A.1.

Primitive constraints can be used as arguments in logical and conditional constraints.

3.2 Logical and conditional constraints

Logical and conditional constraints use primitive constraints as arguments. In addition, JaCoP allows specification of the reified constraints. For detailed list of these constraints see appendix A.4.

3.3 Global Constraints

3.3.1 Alldifferent constraints

Alldifferent constraint ensures that all FDVs from a given list have different values assigned. This constraint uses a simple consistency technique that removes a value, which is assigned to a given FDV from the domains of the other FDVs.

For example, a constraint

   IntVar a = new IntVar(store, "a", 1, 3); 
   IntVar b = new IntVar(store, "b", 1, 3); 
   IntVar c = new IntVar(store, "c", 1, 3); 
   IntVar[] v = {a, b, c}; 
   Constraint ctr = new Alldifferent(v); 
   store.impose(ctr);

enforces that the FDVs a, b, and c have different values.

Alldifferent constraint is provided as three different implementations. Constraint Alldifferent uses a simple implementation described above, i.e., if the domain of a finite domain variable gets assigned a value, the propagation algorithm will remove this value from the other variables. Constraint Alldiff implements this basic pruning method and, in addition, bounds consistency [16]. Finally, constraint Alldistinct implements a generalized arc consistency as proposed by Régin [17].

The example below illustrates the difference in constraints pruning power for Alldifferent and Alldiff constraints. Assume the following variables:

   IntVar a = new IntVar(store, "a", 2, 3); 
   IntVar b = new IntVar(store, "b", 2, 3); 
   IntVar c = new IntVar(store, "c", 1, 3); 
   IntVar[] v = {a, b, c};

The constraints will produce the following results after consistency enforcement.

   store.impose( new Alldifferent(v) ); 
 
   a :: {2..3}, b :: {2..3}, c :: {1..3}

and

   store.impose( new Alldiff(v) ); 
 
   a :: {2..3}, b :: {2..3}, c = 1

Alldistinct constraint will prune domains of variables a, b and c in the same way as Alldiff constraints but, in addition, it can remove single inconsistent values as illustrated below. Assume the following domains for a, b and c.

   a :: {1,3}, b :: {1,3}, c :: {1..3}

The constraints will produce the following results after consistency enforcement.

   store.impose( new Alldistinct(v) ); 
 
   a :: {1,3}, b :: {1,3}, c = 2

3.3.2 Circuit constraint

Circuit constraint tries to enforce that FDVs which represent a directed graph will create a Hamiltonian circuit. The graph is represented by the FDV domains in the following way. Nodes of the graph are numbered from 1 to N. Each position in the list defines a node number. Each FDV domain represents a direct successors of this node. For example, if FDV x at position 2 in the list has domain 1, 3, 4 then nodes 1, 3 and 4 are successors of node x. Finally, if the i’th FDV of the list has value j then there is an arc from i to j.

For example, the constraint

   IntVar a = new IntVar(store, "a", 1, 3); 
   IntVar b = new IntVar(store, "b", 1, 3); 
   IntVar c = new IntVar(store, "c", 1, 3); 
   IntVar[] v = {a, b, c}; 
   Constraint ctr = new Circuit(store, v); 
   store.impose(ctr);

can find a Hamiltonian circuit [2, 3, 1], meaning that node 1 is connected to 2, 2 to 3 and finally, 3 to 1.

3.3.3 Subcircuit constraint

Subcircuit constraint tries to enforce that FDVs which represent a directed graph will create a subcircuit. The graph is represented by the FDV domains in the same way as for the Circuit constraint. The result defines a subcircuit represented by values assigned to the FDVs. Nodes that do not belong to a subcircuit have the value pointing to itself.

For example, the constraint

   IntVar a = new IntVar(store, "a", 1, 3); 
   IntVar b = new IntVar(store, "b", 1, 3); 
   IntVar c = new IntVar(store, "c", 1, 3); 
   IntVar[] v = {a, b, c}; 
   Constraint ctr = new Subcircuit(v); 
   store.impose(ctr);

can find a circuit [2, 1, 3], meaning that node 1 is connected to 2 and 2 to 1 while node 3 is not in the subcircuit and has a value 3.

All solutions to this constraint are [1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1], where the first solution represents a solution with no subsircuits and subseqent solutions define found subcrircuits.

3.3.4 Element constraint

Element constraint of the form Element(I, List, V ) enforces a finite relation between I and V , V = List[I]. The vector of values, List, defines this finite relation. For example, the constraint

   IntVar i = new IntVar(store, "i", 1, 3); 
   IntVar v = new IntVar(store, "v", 1, 50); 
   int[] el = {3, 44, 10}; 
   Constraint ctr = new Element(i, el, v) ; 
   store.impose(ctr);

imposes the relation on the index variable i :: {1..3}, and the value variable v. The initial domain of v will be pruned to v :: {3,10,44} after the intial consistency execution of this Element constraint. The change of one FDV propagates to another FDV. Imposing the constraint V < 44 results in change of I :: {1,3}.

This constraint can be used, for example, to define discrete cost functions of one variable or a relation between task delay and its implementation resources. The constraint is simply implemented as a program which finds values allowed both for the first FDV and third FDV and updating them respectively.

There are also two other versions of element constraint that implement a version of bounds consistency, ElementIntegerFast and ElementVariableFast. This constraints make faster computations but may not remove all inconsistent values.

3.3.5 Distance constraint

Distance constraint computes the absolute value between two FDVs. The result is another FDV, i.e., d = |x - y|.

The example below

   IntVar a = new IntVar(store, "a", 1, 10); 
   IntVar b = new IntVar(store, "b", 2, 4); 
   IntVar c = new IntVar(store, "c", 0, 2); 
   Constraint ctr = new Distance(a, b, c); 
   store.impose(ctr);

produces result a::1..6, b::2..4, c::0..2 since a must be pruned to have distance lower than three.

3.3.6 Cumulative constraint

Cumulative constraint was originally introduced to specify the requirements on tasks which needed to be scheduled on a number of resources. It expresses the fact that at any time instant the total use of these resources for the tasks does not exceed a given limit. It has, in our implementation, four parameters: a list of tasks’ starts Oi, a list of tasks’ durations Di, a list of amount of resources ARi required by each task, and the upper limit of the amount of used resources Limit. All parameters can be either domain variables or integers. The constraint is specified as follows.

   IntVar[] o = {O1, ..., On}; 
   IntVar[] d = {D1, ..., Dn}; 
   IntVar[] r = {AR1, ..., ARn}; 
   IntVar Limit = new IntVar(Store, "limit", 0, 10); 
   Constraint ctr = Cumulative(o, d, r, Limit)

Formally, it enforces the following constraint:

                                 ∑
∀t ∈ [1 m≤iin≤n(Oi),1 m≤aix≤n(Oi + Di)] :       ARk ≤ Limit
                            k:Ok≤t≤Ok+Dk
(3.1)

In the above formulation, min and max stand for the minimum and the maximum values in the domain of each FDV respectively. The constraints ensures that at each time point, t, between the start of the first task (task selected by min(Oi)) and the end of the last task (task selected by max(Oi + Di)) the cumulative resource use by all tasks, k, running at this time point is not greater than the available resource limit. This is shown in Fig 3.1.

JaCoP additionally requires that there exist at least one time point where the number of used resources is equal to Limit, i.e.

                                 ∑
∃t ∈ [ min (Oi), max (Oi + Di)] :       ARk = Limit
    1≤i≤n    1≤i≤n          k:Ok≤t≤Ok+Dk
(3.2)


PIC

Figure 3.1: An example of the profile creation for the cumulative constraint.


There is a version of Cumulative that only requires that the sum of ARk Limit and it is defined with addtional flag as follows. In this constraint, first two flags specify to make propagation based on edge finding algorithm and profile while the last flag informs the solver to not apply reuirement 3.2.

   Constraint ctr = Cumulative(o, d, r, Limit, true, true, false)

3.3.7 Diff2 constraint

Diff2 constraint takes as an argument a list of 2-dimensional rectangles and assures that for each pair i,j (ij) of such rectangles, there exists at least one dimension k where i is after j or j is after i, i.e., the rectangles do not overlap. The rectangle is defined by a 4-tuple [O1, O2, L1, L2], where Oi and Li are respectively called the origin and the length in i-th dimension. The diff2 constraint is specified as follows.

   IntVar[][] rectangles = {{O11, O12, L11, L12}, ..., 
                              {On1, On2, Ln1, Ln2}}; 
   Constraint ctr = new Diff2(rectangles)

The Diff2 constraint can be used to express requirements for packing and placement problems as well as define constraints for scheduling and resource assignment.

This constraint uses two different propagators. The first one tries to narrow Oi and Li FDV’s of each rectangle so that rectangles do not overlap. The second one is similar to the cumulative profile propagator but it is applied in both directions (in 2-dimensional space) for all rectangles. In addition, the constraint checks whether there is enough space to place all rectangles in the limits defined by each rectangle FDV’s domains.

3.3.8 Min, Max and ArgMin, ArgMax constraints

These constraints enforce that a given FDV is minimal or maximal of all variables present on a defined list of FDVs.

For example, a constraint

   IntVar a = new IntVar(Store, "a", 1, 3); 
   IntVar b = new IntVar(Store, "b", 1, 3); 
   IntVar c = new IntVar(Store, "c", 1, 3); 
   IntVar min = new IntVar(Store, "min", 1, 3); 
   IntVar[] v = {a, b, c}; 
   Constraint ctr = new Min(v, min); 
   Store.impose(ctr);

will constraint FDV min to a minimal value of variables a, b and c.

NOTE! The position for parameters in constraints Min and Max is changed comparing to previous versions (i.e., parameters are swapped).

The related constraints ArgMinArgMax find an index of the minimal and maximal element respectively. In the standard version the first position of the minimal or maximal element is constrained. Indexing starts from 1 if the parameter offset is not used.

3.3.9 SumInt, SumBool and LinearInt constraints

These constraints enforce a (wighted) sum of elements of FDVs’ vector

SumInt and SumBool constraint is defined as follows x1 + x2 + ⋅⋅⋅ + xn  sum, where all xi and sum are FDVs and ℜ∈{<,,>,,=,}. SumBool has additional requirement that all FDVs must have domain 0..1.

The weighted sum is provided by LinearInt constraint defined as follows w1 x1 + w2 x2 + ⋅⋅⋅ + wn xn  sum, where all wi and sum are integer constants, xi are FDVs and ℜ∈{<,,>,,=,}.

To define, for example, constraint w1 x1 + w2 x2 + ⋅⋅⋅ + wn xn = s where s is FDV, the following constraint can be used w1 x1 + w2 x2 + ⋅⋅⋅ + (-1) s = 0.

For example, the constraint

   IntVar a = new IntVar(Store, "a", 1, 3); 
   IntVar b = new IntVar(Store, "b", 1, 3); 
   IntVar c = new IntVar(Store, "c", 1, 3); 
   IntVar sum = new IntVar(Store, "sum", 1, 10); 
   IntVar[] v = {a, b, c}; 
   Constraint ctr = new SumInt(v, "==", sum); 
   Store.impose(ctr);

will constraint FDV sum to the sum of a, b and c.

Constraints SumInt, SumBool and LinearInt are primitive constraints. It means that they can be an argument in other constraints, such as Reified, conditional and logical constraints.

For example, the linear constraint can be used in reification, as follows.

   Store store = new Store(); 
   IntVar a = new IntVar(Store, "a", 1, 3); 
   IntVar b = new IntVar(Store, "b", 1, 3); 
   IntVar c = new IntVar(Store, "c", 1, 3); 
   IntVar[] v = {a, b, c}; 
   PrimitiveConstraint ctr = 
      new LinearInt(store, v, new int[] {2, 3, -1}, "<", 10); 
   BooleanVar b = new BooleanVar(store); 
   store.impose(new Reified(ctr), b);

Primitive constraints LinearInt and SumInt implement pruning methods for large linear constraints presented in [9]. There exist also constraint LinearIntDom that implements domain consistency for linear constraint. This constraint has exponential complexity in worst case and need to be used carefully. This constraint can be useful when computing an index from two dimensional array to a vector for element constraint.

Constraints Sum, SumWeight and SumWeightDom extend class Constraint and are deprecated.

3.3.10 ExtensionalSupport and ExtensionalConflict constraints

There exist several implementation of these constraint distinguished by their suffixes. The base implementation with suffix VA tries to balance the usage of memory versus time efficiency. ExtensionalSupportMDD uses multi-valued decision diagram (MDD) as internal representation and algorithms proposed in [4] while ExtensionalSupportSTR uses simple tabular reduction (STR) and the method proposed in [13].

Extensional support and extensional conflict constraints define relations between n FDVs. Both constraints are defined by a vector of n FDVs and a vector of n-tuples of integer values. The n-tuples define the relation between variables defined in the first vector.

The tuples of extensional support constraint define all combinations of values that can be assigned to variables specified in the vector of FDVs. Extensional conflict, on the other hand specifies the combinations of values that are not allowed in any assignment to variables.

The example below specifies the XOR logical relation of the form a b = c using both constraints.

   IntVar a = new IntVar(store, "a", 0, 1); 
   IntVar b = new IntVar(store, "b", 0, 1); 
   IntVar c = new IntVar(store, "c", 0, 1); 
   IntVar[] v = {a, b, c}; 
   // version with ExtensionalSupport constraint 
   store.impose(new ExtensionalSupportVA(v, 
                                         new int[][] {{0, 0, 0}, 
                                                      {0, 1, 1}, 
                                                      {1, 0, 1}, 
                                                      {1, 1, 0}}));
   // version with ExtensionalConflict constraint 
   store.impose(new ExtensionalConflictVA(v, 
                                          new int[][] {{0, 0, 1}, 
                                                       {0, 1, 0}, 
                                                       {1, 0, 0}, 
                                                       {1, 1, 1}}));

3.3.11 Assignment constraint

Assignment constraint implements the following relation between two vectors of FDVs Xi = j Y j = i.

For example, the constraint

   IntVar[] x = new IntVar[4]; 
   IntVar[] y = new IntVar[4]; 
   for (int i=0; i<4; i++) { 
     x[i] = new IntVar(store, "x"+i, 0, 3); 
     y[i] = new IntVar(store, "y"+i, 0, 3); 
   } 
   store.impose(new Assignment(x, y));

produces the following assignment to FDVs when x[1] = 3.

x0::{0..2} 
x1=3 
x2::{0..2} 
x3::{0..2} 
y0::{0, 2..3} 
y1::{0, 2..3} 
y2::{0, 2..3} 
y3=1

The constraint has possibility to define the minimal index for vectors of FDVs. Therefore constraint Assignment(x, y, 1) will index variables from 1 instead of default value 0.

3.3.12 Count constraint

Count constraint counts number of occurrences of value Val on the list List in FDV Var.

For example, the constraint

   IntVar[] List = new IntVar[4]; 
   List[0] = new IntVar(store, "List_"+0, 0, 1); 
   List[1] = new IntVar(store, "List_"+1, 0, 2); 
   List[2] = new IntVar(store, "List_"+2, 2, 2); 
   List[3] = new IntVar(store, "List_"+3, 3, 4); 
 
   IntVar Var = new IntVar(store, "Var", 0, 4): 
 
   store.impose(new Count(List, Var, 2));

produces Var :: {1..2}.

If variable Var will be constrained to 1 then JaCoP will produce produces List_1 :: {0..1} .

NOTE! The position for parameters in constraint Count is changed comparing to previous versions.

3.3.13 Values constraint

Values constraint takes as arguments a list of variables and a counting variable. It counts a number of different values on the list of variables in the counting variable. For example, consider the following code.

   IntVar x0 = new IntVar(store, "x0", 1,1); 
   IntVar x1 = new IntVar(store, "x1", 1,1); 
   IntVar x2 = new IntVar(store, "x2", 3,3); 
   IntVar x3 = new IntVar(store, "x3", 1,3); 
   IntVar x4 = new IntVar(store, "x4", 1,1); 
   IntVar[] val = {x0, x1, x2, x3, x4}; 
   IntVar count = new IntVar(store, "count", 2, 2); 
   store.impose( new Values(count, val) );

Constraint Values will remove value 2 from variable x3 to assure that are only two different values (1 and 3) on the list of variables as specified by variable count.

NOTE! The position for parameters in constraint Values is changed comparing to previous versions.

3.3.14 Global cardinality constraint (GCC)

Global cardinality constraint (GCC) is defined using two lists of variables. The first list is the value list and the second list is the counter list. The constraint counts number of occurrences of different values in the variables from the value list. The counter list is used to counter occurrences of a specific value. It can also specify the number of allowed occurrences of a specific value on the value list. Variables on the counter list are assigned to values as follows. The lowest value in the domain of all variables from the value list is counted by the first variable on the counters list. The next value (+1) is counted by the next variable and so on.

For example, the following code counts number of values -1, 0, 1 and 2 on value list x. The values are counted using counter list y using the following mapping. -1 is counted in y0, 0 is counted in y1, 1 is counted in y2 and 2 is counted in y3.

   IntVar x0 = new IntVar(store, "x0", -1, 2); 
   IntVar x1 = new IntVar(store, "x1", 0, 2); 
   IntVar x2 = new IntVar(store, "x2", 0, 2); 
   IntVar[] x = {x0, x1, x2}; 
 
   IntVar y0 = new IntVar(store, "y0", 1, 1); 
   IntVar y1 = new IntVar(store, "y1", 0, 1); 
   IntVar y2 = new IntVar(store, "y2", 0, 1); 
   IntVar y3 = new IntVar(store, "y3", 1, 2); 
   IntVar[] y = {y0, y1, y2, y3}; 
 
   store.impose(new GCC(x, y));

The GCC constraint will allow only the following five combinations of x variables [x0=-1, x1=0, x2=2], [x0=-1, x1=1, x2=2], [x0=-1, x1=2, x2=0], [x0=-1, x1=2, x2=1], and [x0=-1, x1=2, x2=2].

3.3.15 Among and AmongVar

Among constraint is specified using three parameters. The first parameter is the value list, the second one is a set of values specified as IntervalDomain, and finally the third parameter, the counter, counts the number of variables from the value list that get assigned values from the set of values. The constraint assures that exactly the number of variables defined by count variable is equal to one value from the set of values.

The following example constraints that either 2 or 4 variables from value list numbers are equal either 1 or 3. There exist 2880 such assignments.

   IntVar numbers[] = new IntVar[5]; 
   for (int i = 0; i < numbers.length; i++) 
      numbers[i] = new IntVar(store, "n" + i, 0,5); 
 
   IntVar count = new IntVar(store, "count", 2,2); 
   count.addDom(4,4); 
   IntervalDomain val = new IntervalDomain(1,1); 
   val.addDom(3,3); 
   store.impose(new Among(numbers, val, count));

AmongVar constraint is a generalization of Among constraint. Instead of specifying a set of values it uses a list of variables as the second parameter. It counts how many variables from the value list are equal to at least one variable from list of variables (second parameter).

The example below specifies the same conditions as the Among constraint in the above example.

   IntVar numbers[] = new IntVar[5]; 
   for (int i = 0; i < numbers.length; i++) 
      numbers[i] = new IntVar(store, "n" + i, 0,5); 
 
   IntVar count = new IntVar(store, "count", 2,2); 
   count.addDom(4,4); 
   IntVar[] values = new IntVar[2]; 
   values[0] = new IntVar(store, 1,1); 
   values[1] = new IntVar(store, 3,3); 
   store.impose(new AmongVar(numbers, values, count));

3.3.16 Regular constraint

Regular constraint accepts only the assignment to variables that are accepted by an automaton. The automaton is specified as the first parameter of this constraint and a list of variable is the second parameter. This constraint implements a polynomial algorithm to establish GAC.

The automaton is specified by its states and transitions. There are three types of states: initial state, intermediate states, and final states. Each transition has associated domain containing all values which can trigger this transition. Values assigned to transitions must be present in the domains of assigned constraint variable. Each value may cause firing of the related transition. The automaton eventually reaches a final state after taking the last transition as specified by the value of the last variable.

Each state can be assigned a level by topologically sorting states of the automaton. The variables from the list (second parameters) are assigned to these levels. All states at the same level are assigned the same variable (see Figure 3.2). If necessary, the automaton, containing cycles, is unrolled to match a list of variables. Each transitions has assigned values that are allowed for a variable when the transition in the automaton is selected. This is specified as the interval domain.

The example below implements the automaton from Figure 3.2. This automaton defines condition for three variables to be different values 0, 1 or 2.


PIC

Figure 3.2: An example of the automaton for Regular constraint.


   IntVar[] var = new IntVar[3]; 
   var[0] = new IntVar(store, "v"+0, 0, 2); 
   var[1] = new IntVar(store, "v"+1, 0, 2); 
   var[2] = new IntVar(store, "v"+2, 0, 2); 
 
   FSM g = new FSM(); 
   FSMState[] s = new FSMState[8]; 
   for (int i=0; i<s.length; i++) { 
      s[i] = new FSMState(); 
      g.allStates.add(s[i]); 
   } 
   g.initState = s[0]; 
   g.finalStates.add(s[7]); 
 
   s[0].transitions.add(new FSMTransition(new IntervalDomain(0, 0), 
                                          s[1])); 
   s[0].transitions.add(new FSMTransition(new IntervalDomain(1, 1), 
                                          s[2])); 
   s[0].transitions.add(new FSMTransition(new IntervalDomain(2, 2), 
                                          s[3])); 
 
   s[1].transitions.add(new FSMTransition(new IntervalDomain(1, 1), 
                                          s[4])); 
   s[1].transitions.add(new FSMTransition(new IntervalDomain(2,2), 
                                          s[5])); 
 
   s[2].transitions.add(new FSMTransition(new IntervalDomain(0, 0), 
                                          s[4])); 
   s[2].transitions.add(new FSMTransition(new IntervalDomain(2,2), 
                                          s[6])); 
 
   s[3].transitions.add(new FSMTransition(new IntervalDomain(0, 0), 
                                          s[5])); 
   s[3].transitions.add(new FSMTransition(new IntervalDomain(1, 1), 
                                          s[6])); 
 
   s[4].transitions.add(new FSMTransition(new IntervalDomain(2, 2), 
                                          s[7])); 
   s[5].transitions.add(new FSMTransition(new IntervalDomain(1, 1), 
                                          s[7])); 
   s[6].transitions.add(new FSMTransition(new IntervalDomain(0, 0), 
                                          s[7])); 
 
   store.impose(new Regular(g, var));

3.3.17 Knapsack constraint

Knapsack constraint specifies knapsack problem. This implementation1 was inspired by the paper [11] and published in [14]. The major extensions of that paper are the following. The quantity variables do not have to be binary. The profit and capacity of the knapsacks do not have to be integers. In both cases, the constraint accepts any finite domain variable.

The constraint specify number of categories of items. Each item has a given weight and profit. Both weight and profit are specified as positive integers. The problem is to select a number of items in each category to satisfy capacity constraint, i.e. the total weight must be in the limits specified by the capacity variable. Each such solution is then characterized by a given profit. It is defined in JaCoP as follows.

Knapsack(int[] profits, int[] weights, IntVar[] quantity, 
         IntVar knapsackCapacity, IntVar knapsackProfit)

It can be formalize using the following constraints.

∑
    weights[i]⋅quantity[i] = knapsackCapacity
∑ i
    profits[i]⋅quantity[i] = knapsackP rofit
  i
∀i : weights[i] ∈ ℤ >0 ∧ ∀i : profits[i] ∈ ℤ >0
(3.3)

(3.4)

(3.5)

3.3.18 Geost constraint

Geost is a geometrical constraint, which means that it applies to geometrical objects. It models placement problems under geometrical constraints, such as non overlapping constraints. Geost consistency algorithm was proposed by Beldiceanu et al [18]. The implementation of Geost in JaCoP is a result of a master thesis by Marc-Olivier Fleury.

In order to describe the constraint, we will introduce several definitions and relate them to JaCoP implementation.

Definition 1 A shifted box b is a pair (b.t[],b.l[]) of vectors of integers of length k, where k is the number of dimensions of the problem. The origin of the box relative to a given reference is b.t[], and b.l[] contains the length of the box, for each dimension.


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Figure 3.3: Shifted box in 2 dimensions. Reference origin is denoted by the black square.


Shifted box is defined in JaCoP using class DBox. For example, a two dimensional shifted box starting at coordinates (0,0) and having length 2 in first dimension and 1 in second direction is specified as follows.

    DBox sbox = new DBox(new int[] {0,0}, new int[] {2,1});

Definition 2 A shape s is a set of shifted boxes. It has a unique identifier s.sid.


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Figure 3.4: Example of a shape in 2 dimensions. Reference origin is denoted by the black square.


In JaCoP, we can define n shapes as collection of shifted boxes. Shape identifiers start at 0 and must be assigned consecutive integers. Therefore we have shapes with identifiers in interval 0..n-1. The following JaCoP example defines a shape with identifier 0, consisting of three sboxs, depicted in Figure 3.4.

    ArrayList<Shape> shapes = new ArrayList<Shape>(); 
 
    ArrayList<DBox> shape1 = new ArrayList<DBox>(); 
    shape1.add(new DBox(new int[] {0,0}, new int[] {2,1})); 
    shape1.add(new DBox(new int[] {0,1}, new int[] {1,2})); 
    shape1.add(new DBox(new int[] {1,2}, new int[] {3,1})); 
 
    shapes.add( new Shape(0, shape1) );

Definition 3 An object o is a tuple (o.id,o.sid,o.x[], o.start,o.duration,o.end). o.id is a unique identifier, o.sid is a variable that stores all shapes that o can take. o.x[] is a k-dimensional vector of variables which represent the origin of o. o.start, o.duration and o.end define the interval of time in which o is present.

An object in JaCoP is defined by class GeostObject. It specifies basically all parameters of an object. An example below specifies object 0 that can take shapes 0, 1, 2 and 3. The object can be placed using coordinates (X_o1, Y_o1). The object is present during time 2 to 14.

    ArrayList<GeostObject> objects = new ArrayList<GeostObject>(); 
 
    IntVar X_o1 = new IntVar(store, "x1", 0, 5); 
    IntVar Y_o1 = new IntVar(store, "y1", 0, 5); 
    IntVar[] coords_o1 = {X_o1, Y_o1}; 
    IntVar shape_o1 = new IntVar(store, "shape_o1", 0, 3); 
    IntVar start_o1 = new IntVar(store, "start_o1", 2, 2); 
    IntVar duration_o1 = new IntVar(store, "duration_o1", 12, 12); 
    IntVar end_o1 = new IntVar(store, "end_o1", 14, 14); 
    GeostObject o1 = new GeostObject(0, coords_o1, shape_o1, 
                                     start_o1, duration_o1, end_o1); 
    objects.add(o1);

Note that since object shapes are defined in terms of collections of shifted boxes, and since shifted boxes have a fixed size, Geost is not suited to solve problems in which object sizes can vary. Polymorphism provides some flexibility (shape variable having multiple values in their domain), but it is essentially intended to allow the modeling of objects that can take a small amount of different shapes. Typically objects that can be rotated. The duration of an object can be useful in cases where objects have variable sizes, because it is a variable, which means that some more flexibility is available. However, this feature is only available for one dimension. These restrictions are design choices made by the authors of Geost, probably because it fits well their primary field of application, which consists in packing goods in trucks. Using fixed sized shapes is also useful because it allows more deductions concerning possible placements.

When all shapes and objects are defined it is possible to specify geometrical constraints that must be fulfilled when placing these objects. Implemented geometrical constraints include in-area and non-overlapping constraints. In-area constraint enforces that objects have to lie inside a given k-dimensional sbox. Non-overlapping constraints require that no two objects can overlap.

The code below specifies two geometrical constraint, non-overlapping and in-area. They are specified by classes NonOverlapping and InArea. It must be noted that non-overlapping constraint in the code below specifies that all objects must not overlap in its two dimensions and time dimension (the time dimension is implemented as one additional dimension and therefore we specify dimensions 0, 1 and 2). In-area constraint requires that all object must be included in the sbox of dimensions 5x4.

    ArrayList<ExternalConstraint> constraints = 
                    new ArrayList<ExternalConstraint>(); 
    int[] dimensions = {0, 1, 2}; 
    NonOverlapping constraint1 = 
                    new NonOverlapping(objects, dimensions); 
    constraints.add(constraint1); 
    InArea constraint2 = new InArea( 
                  new DBox(new int[] {0,0}, new int[] {5,4}), null); 
    constraints.add(constraint2);

Finally, the Geost constraint is imposed using the following code.

    store.impose( new Geost(objects, constraints, shapes) );

3.3.19 NetworkFlow constraint

NetworkFlow constraint defines a minimum-cost network flow problem. An instance of this problem is defined on a directed graph by a tuple (N,A,l,u,c,b), where

A flow is a function x : A 0. The minimum-cost flow problem asks to find a flow that satisfies all arc capacity and node balance conditions, while minimizing total cost. It can be stated as follows:

min   z(x) = (i,j)Acijxij (3.6)
s.t.   lij xij uij (i,j) A, (3.7)
j:(i,j)Axij - j:(j,i)Axji = bi i N (3.8)

The network is built with NetworkBuilder class using node, defined by class org.jacop.net.Node and arcs. Each node is defined by its name and node mass balance b. For example, node A with balance 0 is defined using network net as follows.

    NetworkBuilder net = new NetworkBuilder(); 
    Node A = net.addNode("A", 0);

Source node producing flow of capacity 5 and sink node consuming flow of capacity 5 are defined using similar constructs but different value of node mass balance, as indicated below.

      Node source = net.addNode("source", 5); 
      Node sink = net.addNode("sink", -5);

Arc of the network are defined always between two nodes. They define connection between given nodes, the lower and upper capacity values assigned to the arc (values l and u) as well as the flow cost-per-unit function on the arc (value c). This can be defined using different methods with either integers or finite domain variables. For example, an arc from source to node A with l = 0 and u = 5 and c = 3 can be defined as follows.

      net.addArc(A, B, 3, 0, 5);

or

      x = new IntVar(store, "source->A", 0, 5); 
      net.addArc(A, B, 3, x);

It can be noted, that cost-per-unit value can also be defined as a finite domain variable.

The constraint has also the flow cost, defined as z(x) in 3.6. It is defined as follows.

     IntVar cost = new IntVar(store, "cost", 0, 1000); 
     net.setCostVariable(cost);

Note that the NetworkFlow only ensures that cost z(x) Zmax, where z(x) is the total cost of the flow (see equation (3.6)). In our code it is defined as variable cost.

The constraint is finally posed using the following method.

      store.impose(new NetworkFlow(net));

For example, Figure 3.5 presents the code for network flow problem depicted in Figure 3.6. The minimal flow, found by the solver, is 10 that is indicated in the figure.


      store = new Store(); 
 
      IntVar[] x = new IntVar[8]; 
 
      NetworkBuilder net = new NetworkBuilder(); 
      Node source = net.addNode("source", 5); 
      Node sink = net.addNode("sink", -5); 
 
      Node A = net.addNode("A", 0); 
      Node B = net.addNode("B", 0); 
      Node C = net.addNode("C", 0); 
      Node D = net.addNode("D", 0); 
 
      x[0] = new IntVar(store, "x_0", 0, 5); 
      x[1] = new IntVar(store, "x_1", 0, 5); 
      net.addArc(source, A, 0, x[0]); 
      net.addArc(source, C, 0, x[1]); 
 
      x[2] = new IntVar(store, "a->b", 0, 5); 
      x[3] = new IntVar(store, "a->d", 0, 5); 
      x[4] = new IntVar(store, "c->b", 0, 5); 
      x[5] = new IntVar(store, "c->d", 0, 5); 
      net.addArc(A, B, 3, x[2]); 
      net.addArc(A, D, 2, x[3]); 
      net.addArc(C, B, 5, x[4]); 
      net.addArc(C, D, 6, x[5]); 
 
      x[6] = new IntVar(store, "x_6", 0, 5); 
      x[7] = new IntVar(store, "x_7", 0, 5); 
      net.addArc(B, sink, 0, x[6]); 
      net.addArc(D, sink, 0, x[7]); 
 
      IntVar cost = new IntVar(store, "cost", 0, 1000); 
      net.setCostVariable(cost); 
 
      store.impose(new NetworkFlow(net));

Figure 3.5: Constraint for network flow model from Figure 3.6



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Figure 3.6: An example network and its minimal flow.


Network builder has a special attribute handlerList that makes it possible to specify structural rules connected to the network. Each structural rule must implement VarHandler interface, which allows network flow constraint to cooperate with the structural rule. An important, already implemented rule, is available in the class DomainStructure. It specifies, for each structural variable sv, a list of arcs that the structural rule influence. This structural rule makes it possible to enforce minimum or maximum amount of flow on a given arc depending on the relationship between the domain of variable sv and domain d, specified within a structural rule. The domain of variable sv and domain d do not intersect if and only if the flow on a given arc is minimal as specified by initial value x.min(), denoted as xmin. Moreover, the domain of varibale sv is contained within domain d if and only if the actual flow on a given arc is maximal as specified by initial value x.max(), denoted as xmax. It is enforced by the following rules.

sv.dom () ⊆ d  ⇒  x = xmax,
   x < xmax  ⇒  sv.dom()∩ d = ∅,
(3.9)

sv.dom () ∩d = ∅  ⇒   x = xmin,
       x > xmin ⇒   sv.dom () ⊆ d,
(3.10)

For example, creation of structural rule for arc between node source and B will enforce that this arc will have maximal flow if variable s is zero and minimal flow otherwise. The rule works also in the other direction, i.e. if the flow will be maximal variable s=0 and if the flow will be minimal this variable will be 1.

      Arc[] arcs = new Arc[1]; 
      arcs[0] = net.addArc(source, B, 0, x[1]); 
 
      IntVar s = new IntVar(store, "s", 0, 1); 
      Domain[] domCond = new IntDomain[1]; 
      domCond[0] = new IntervalDomain(0, 0); 
 
      net.handlerList.add(new DomainStructure(s, 
                          Arrays.asList(domCond), 
                          Arrays.asList(arcs)));

SoftAlldifferent constraint as well as SoftGCC constraint use DomainStructure rules to enforce flow from variable nodes to value nodes according the actual domain/value of variables represented by variable nodes. It is crucial functionality for the implementation of those soft global constraints.

3.3.20 Binpacking

Binpacking constraint defines a problem of packing bins with the specified items of a given size. Each bin has a defined load. The constraint is defined as follows.

Binpacking(IntVar[] item, IntVar[] load, int[] size)

where item means the bin number assigned to item at position i, load defines load of bin i as finite domain variable (min and max load) and size defines items i size.

It can be formalize using the following formulation.

       ∑
∀i            size[j] = load[i]
  j whereitem[j]=i
(3.11)

The binpacking constraint implements methods and algorithms proposed in [20].

3.3.21 LexOrder

The LexOrder constraint enforces ascending lexicographic order between two vectors that can be of different size. The constraints makes it possible to enforce strict ascending lexicographic order, that is the fist vector must be always before the seconf one in the lexicographical order, or it can allow equality between the vectors. This is controlled by the last parameter of the constraint that is either true for strictly ascending lexicographic order or false otherwise. The default is non-strictly ascending lexicographic order when no second parameter is defined.

The following code specifies Lex constraint that holds when the strict ascending lexicographic order holds between two vectors of 0/1 varables.

      int N = 2; 
      IntVar[] x = new IntVar[N]; 
      for (int i = 0; i < x.length; i++) 
          x[i] = new IntVar(store, "x["+i+"]", 0, 1); 
 
      IntVar[] y = new IntVar[N]; 
      for (int i = 0; i < y.length; i++) 
          y[i] = new IntVar(store, "y["+i+"]", 0, 1); 
 
      store.impose(new LexOrder(x, y, true));

The result is as follows.

[0, 0, 0], [0, 0, 1] 
[0, 0, 0], [0, 1, 0] 
[0, 0, 0], [0, 1, 1] 
[0, 0, 0], [1, 0, 0] 
[0, 0, 0], [1, 0, 1] 
[0, 0, 0], [1, 1, 0] 
[0, 0, 0], [1, 1, 1] 
[0, 0, 1], [0, 1, 0] 
[0, 0, 1], [0, 1, 1] 
[0, 0, 1], [1, 0, 0] 
[0, 0, 1], [1, 0, 1] 
[0, 0, 1], [1, 1, 0] 
[0, 0, 1], [1, 1, 1] 
[0, 1, 0], [0, 1, 1] 
[0, 1, 0], [1, 0, 0] 
[0, 1, 0], [1, 0, 1] 
[0, 1, 0], [1, 1, 0] 
[0, 1, 0], [1, 1, 1] 
[0, 1, 1], [1, 0, 0] 
[0, 1, 1], [1, 0, 1] 
[0, 1, 1], [1, 1, 0] 
[0, 1, 1], [1, 1, 1] 
[1, 0, 0], [1, 0, 1] 
[1, 0, 0], [1, 1, 0] 
[1, 0, 0], [1, 1, 1] 
[1, 0, 1], [1, 1, 0] 
[1, 0, 1], [1, 1, 1] 
[1, 1, 0], [1, 1, 1]

The algorithm of this constraint is based on the algorithm presented in [7].

3.4 Decomposed constraints

Decomposed constraints do not define any new constraints and related pruning algorithms. They are translated into existing JaCoP constraints. Sequence and Stretch constraints are decomposed using Regular constraint.

Decomposed constraints are imposed using imposeDecomposition method instead of ordinary impose method.

3.4.1 Sequence constraint

Sequence constraint restricts values assigned to variables from a list of variables in such a way that any sub-sequence of length q contains N values from a specified set of values. Value N is further restricted by specifying min and max allowed values. Value q, min and max must be integer.

The following code defines restrictions for a list of five variables. Each sub-sequence of size 3 must contain 2 (min = 2 and max = 2) values 1.

   IntVar[] var = new IntVar[5]; 
   for (int i=0; i<var.length; i++) 
      var[i] = new IntVar(store, "v"+i, 0, 2); 
 
   store.imposeDecomposition(new Sequence(var, //variable list 
                        new IntervalDomain(1,1), //set of values 
                        3, // q, sequence length 
                        2, // min 
                        2  // max 
                      ));

There exist ten solutions: [01101, 01121, 10110, 10112, 11011, 11211, 12110, 12112, 21101, 21121]

3.4.2 Stretch constraint

Stretch constraint defines what values can be taken by variables from a list and how sub-sequences of these values are formed. For each possible value it specifies a minimum (min) and maximum (max) length of the sub-sequence of these values.

For example, consider a list of five variables that can be assigned values 1 or 2. Moreover we constraint that the sub-sequence of value 1 must have length 1 or 2 and the sub-sequence of value 2 must have length either 2 or 3. The following code specifies these restrictions.

   IntVar[] var = new IntVar[5]; 
   for (int i=0; i<var.length; i++) 
      var[i] = new IntVar(store, "v"+i, 1, 2); 
 
   store.imposeDecomposition( 
                     new Stretch(new int[] {1, 2},  // values 
                                 new int[] {1, 2},  // min for 1 2 
                                 new int[] {2, 3},  // max for 1 2 
                                 var // variables 
                            ));

This program produces six solutions: [11221, 11222, 12211, 12221, 22122, 22211]

3.4.3 Lex constraint

The Lex constraint enforces ascending lexicographic order between n vectors that can be of different size. The constraints makes it possible to enforce strict ascending lexicographic order, that is vector i must be always before vector i + 1 in the lexicographical order, or it can allow equality between consecutive vectors. This is controlled by the last parameter of the constraint that is either true for strictly ascending lexicographic order or false otherwise. The default is non-strictly ascending lexicographic order when no second parameter is defined.

The following code specifies Lex constraint that holds when the strict ascending lexicographic order holds between four vectors of different sizes.

      IntVar[] d = new IntVar[6]; 
      for (int i = 0; i < d.length; i++) { 
          d[i] = new IntVar(store, "d["+i+"]", 0, 2); 
      } 
 
      IntVar[][] x = { {d[0],d[1]}, {d[2]}, {d[3],d[4]}, {d[5]} }; 
 
      store.imposeDecomposition(new Lex(x, true));

This program produces nine following solutions.

[[0, 0], [1], [1, 0], [2]],

[[0, 0], [1], [1, 1], [2]],

[[0, 0], [1], [1, 2], [2]],

[[0, 1], [1], [1, 0], [2]],

[[0, 1], [1], [1, 1], [2]],

[[0, 1], [1], [1, 2], [2]],

[[0, 2], [1], [1, 0], [2]],

[[0, 2], [1], [1, 1], [2]],

[[0, 2], [1], [1, 2], [2]]

3.4.4 Soft-Alldifferent

Soft-alldifferent makes it possible to violate to some degree the alldifferent relation. The violations will come at a cost which is represented by cost variable. This constraint is decomposed with the help of network flow constraint.

There are two violation measures supported, decomposition based, where violation of any inequality relation between any pair contributes one unit of cost to the cost metric. The other violation measure is called variable based, which simply states how many times a variable takes value that is already taken by another variable.

The code below imposes a soft-alldifferent constraint over five variables with cost defined as being between 0 and 20.

   IntVar[] x = new IntVar[5]; 
   for (int i=0; i< x.length; i++) 
      x[i] = new IntVar(store, "x"+i, 1, 4); 
   IntVar cost = new IntVar(store, "cost", 0, 20); 
 
   store.imposeDecomposition(new SoftAlldifferent(x, cost, 
                             ViolationMeasure.DECOMPOSITION_BASED );

3.4.5 Soft-GCC

Soft-GCC constraint makes it possible to violate to some degree GCC constraint. The Soft-GCC constraint requires number of arguments. In the code example below, vars specify the list of variables, which values are being counted, and the list of integers countedValues specifies the values that are counted. Values are counted in two counters specified by a programmer. The first list of counter variables, denoted by hardCounters in our example, specifies the hard limits that can not be violated. The second list, softCounters, specifies preferred values for counters and can be violated. Each position of a variable on these lists corresponds to the position of the value being counted on list countedValues. In practice, domains of variables on list hardCounters should be larger than domains of corresponding variables on list softCounters.

Soft-GCC accepts only value based violation metric that, for each counted value, sums up the shortage or the excess of a given value among vars. There are other constructors of Soft-GCC constraint that allows to specify hard and soft counting constraints in multitude of different ways.

store.imposeDecomposition(new SoftGCC(vars, hardCounters, 
                          countedValues, softCounters, 
                          cost, ViolationMeasure.VALUE_BASED));

Chapter 4
Set Solver

JaCoP provides a library of set constraints in library org.jacop.set. This implementation is based on set intervals and related operations as originally presented in [8] and the first version has been implemented as a diploma project1 . JaCoP definition of set intervals, set domains and set variables is presented in section 4.1. Available constraints are discussed in section 4.2. Search, using set variables, is introduced in section 4.3.

4.1 Set Variables and Set Domains

Set is defined as an ordered collection of integers using class org.jacop.core.IntervalDomain and a set domain as abstract class org.jacop.set.core.SetDomain. Currently, there exist only one implementation of set domain as a set interval, called BoundSetDomain. The set interval for BoundSetDomain d is defined by its greatest lower bound (glb(d)) and its least upper bound (lub(d)). For example, set domain d = {{1}..{1..3}} is defined with glb(d) = {1}, set containing element 1, and lub(d) = {1..3}, set containing elements 1, 2 and 3. This set domain represent a set of sets {{1},{1..2},{1,3},{1..3}}. Each set domain to be correct must have glb(d) lub(d). glb(d) can be considered as a set of all elements that are members of the set and lub(d) specifies the largest possible set.

The following statement defines set variable s for the set domain discussed above.

SetVar s = new SetVar(store, "s", 
      new BoundSetDomain(new IntervalDomain(1,1), 
                         new IntervalDomain(1,3)));

BoundSetDomain can specify a typical set domain, such as d = {{}..{1..3}}, in a simple way as

SetVar s = new SetVar(store, "s", 1, 3);

and an empty set domain as

SetVar s = new SetVar(store, "s", new BoundSetDomain());

Set domain can be created using IntervalDomain and BoundSetDomain class methods. They make it possible to form different sets by adding elements to sets.

4.2 Set Constraints

JaCoP implements a number of set constraints specified in appendix A.2. Constraints AinS, AeqB and AinB are primitive constraints and can be reified and used in other constraints, such as conditional and logical. Other constraints are treated as ordinary JaCoP constraints.

Consider the following code that uses union constraint.

SetVar s1 = new SetVar(store, "s1", 
                new BoundSetDomain(new IntervalDomain(1,1), 
                                   new IntervalDomain(1,4))); 
SetVar s2 = new SetVar(store, "s2", 
                new BoundSetDomain(new IntervalDomain(2,2), 
                                   new IntervalDomain(2,5))); 
SetVar s = new SetVar(store, "s", 1,10); 
Constraint c = new AunionBeqC(s1, s2, s);

It performs operation {{1}..{1..4}} {{2}..{2..5}} = {{}..{1..10}} and produces {{1}..{1..4}} {{2}..{2..5}} = {{1..2}..{1..5}}. This represents 108 possible solutions.

4.3 Set Search

Set variables will require different search organization. Basically, during search the decisions will be made whether an element belongs to a set or it does not belong to this set.

JaCoP still uses DepthFirstSearch but needs different methods for set variable selection implementing ComparatorVariable and a method for value selection implementing Indomain. The special methods are specified in appendix B.2. In addition, variable selection methods MostConstrainedStatic and MostConstrainedDynamic will also work.

An example search can be specified as follows.

   Search<SetVar> search = new DepthFirstSearch<SetVar>(); 
 
   SelectChoicePoint<SetVar> select = new SimpleSelect<SetVar>( 
                                vars, 
                                new MinLubCard<SetVar>(), 
                                new MaxGlbCard<SetVar>(), 
                                new IndomainsetMin<SetVar>()); 
   search.setSolutionListener(new SimpleSolutionListener<SetVar>()); 
 
   boolean result = search.labeling(store, select);

Chapter 5
Floating Point Solver

JaCoP provides a library of floating point constraints in library org.jacop.floats. This implementation is based on floating point intervals and related constraints.

5.1 Floating Point Variables and Floating Point Domains

Floating point domain is defined in a similar way as integer variable domain as an ordered list of floating point intervals in class org.jacop.floats.core.FloatIntervalDomain. This class implements an abstract class org.jacop.floats.core.FloatDomain that in turn extends ordinary JaCoP domain defined in org.jacop.core.Domain. The floating point variable is defined using the floating point interval as follows.

FloatVar f = new FloatVar(store, "f", 0.0, 10.0);

This java declaration defines floating point variable f with domain 0.0..10.0.

The float variables are used in floating point constraints, discussed in the next section. Special attention is paid to precision of floating point operations. Basically, JaCoP considers a floating point interval to represent a single value if a difference between max and min values of a variable is lower than an epsilon value (ϵ). Epsilon is calculated based on a given precision and ulp value (ulp is a floating point specific value know as “unit at last position”). The precision is defined in org.jacop.core.FloatDomain and can be set-up and obtained using following methods FloatDomain.setPrecision(double p) and FloatDomain.precision(). Epsilon is defined as a maximum of ulp and the precision. Class FloatDomain defines also min and max values of floating variables and values of π and e.

5.2 Floating Point Constraints

JaCoP implements a number of floating point constraints specified in appendix A.3.

Constraints PeqC, PeqQ, PgtC, PgtQ, PgteqC, PgteqQ, PltC, PltQ, PlteqC, PlteqQ, PneqC, PneqQ, PplusCeqR,PplusQeqR, PminusCeqR, PminusQeqR and LinearFloat are primitive constraints and can be reified and used in other constraints, such conditional and logical. Other constraints are treated as ordinary JaCoP constraints.

Consider the following code that uses division constraint.

FloatDomain.setPrecision(1E-4); 
 
FloatVar a = new FloatVar(store, "a", 0.1, 3.0); 
FloatVar b = new FloatVar(store, "b", -0.2, 6.0); 
FloatVar c = new FloatVar(store, "c", -1000, 1000); 
 
store.impose(new PdivQeqR(a, b, c));

It performs operation {0.100..3.000} / {-0.200..6.000} and produces a pruned domain for c::{-1000.0000..-0.5000, 0.0167..1000.0000}, assuming precision 10-4.

5.3 Floating Point Search

Floating point variables will require special search definition. Basically, a domain split search (bisection) is used together with consistency checking to find a solution.

JaCoP standard search for floating point variables still uses DepthFirstSearch but needs to use specific methods for selection of choice points as well as variable selection. org.jacop.floats.search.SplitSelectFloat is used to define how intervals of floating point variables will be split. Default behavior of this method is to split an interval in a middle and try the left part first. If variable leftFirst is set to false the right interval will be checked first. The variables will be selected based on the variable selection heuristic defined as a parameter for SplitSelectFloat class. There exist several methods to select variables specific for FloatVar. They are defined in org.jacop.floats.search. Other general methods that are not specific for a given domain can also be used here, for example org.jacop.search.MostConstrainedStatic.

An example search can be specified as follows.

DepthFirstSearch<FloatVar> search = new DepthFirstSearch<FloatVar>(); 
SplitSelectFloat<FloatVar> select = new SplitSelectFloat<FloatVar>( 
                               store, x, 
                               new LargestDomainFloat<FloatVar>()); 
search.setSolutionListener(new PrintOutListener<FloatVar>()); 
 
boolean result = search.labeling(store, select);

Minimization, using DepthFirstSearch class, is done as for other variables by calling labeling method with the third parameter defining cost variable. Both IntVar and FloatVar can be used to defined cost criteria for minimization.

JaCoP offers also another minimization method. This method uses cost variable to lead the minimization. The method splits the domain of cost variable into two parts and then checks, using standard DepthFirstSearch labeling method if there is a solution in this part. If there is a solution in this part it continues to split the current interval and checks the left interval again. If there is no solution in this part (depth-first-search return false) it will try the right interval. The search finishes when the cost variable is ground (the difference between max and min is lower or equal epsilon). An example code is presented below.

DepthFirstSearch<FloatVar> search = new DepthFirstSearch<FloatVar>(); 
SplitSelectFloat<FloatVar> select = new SplitSelectFloat<FloatVar>( 
                           store, x, 
                           new LargestDomainFloat<FloatVar>()); 
 
Optimize min = new Optimize(store, search, select, cost); 
boolean result = min.minimize();

It is also possible to define SplitSelectFloat with variable selection heuristic (the last parameter) equal null. In this case a default method will be used that is round-robin selection. The variables are selected in a round-robin fashion until their domains will not become ground.

5.4 Example

The following code defines equation x2 + sin(1∕x3) = 0 together with the search that finds all 318 solutions.

FloatDomain.setPrecision(1.0e-14); 
FloatDomain.intervalPrint(false); 
Store store = new Store(); 
 
FloatVar x = new FloatVar(store, "x", 0.1, 1.0); 
FloatVar zero = new FloatVar(store, 0,0); 
FloatVar one = new FloatVar(store, 1,1); 
 
// x*x + sin(1.0/(x*x*x)) = 0.0; 
FloatVar[] temp = new FloatVar[4]; 
for (int i=0; i<temp.length; i++) 
    temp[i] = new FloatVar(store, "temp["+i+"]", -1e150, 1e150); 
store.impose(new PmulQeqR(x, x, temp[0])); 
store.impose(new PmulQeqR(x, temp[0], temp[1])); 
store.impose(new PdivQeqR(one, temp[1], temp[2])); 
store.impose(new SinPeqR(temp[2], temp[3])); 
store.impose(new PplusQeqR(temp[0], temp[3], zero)); 
 
DepthFirstSearch<FloatVar> search = new DepthFirstSearch<FloatVar>(); 
SplitSelectFloat<FloatVar> s = new SplitSelectFloat<FloatVar>(store, 
                                   new FloatVar[] {x}, null); 
search.setSolutionListener(new PrintOutListener<FloatVar>()); 
search.getSolutionListener().searchAll(true); 
 
search.labeling(store, s);

The last solution is x = 0.65351684593772.

5.5 Experimental Extensions

JaCoP package org.jacop.floats offers also experimental extensions. They include a method to generate constraints that define the derivative of a function, represented by constraints, and a interval Newton method.

5.5.1 Derivatives

JaCoP can do symbolic (partial) derivatives of functions, that is using the defined constraints for a given function and a variable defining the result of the function, JaCoP will generate constraints that define a function for a derivative of the original function. There is a complication in this process since constraints are not functions and sometime it is difficult or impossible to find out which constraint defines a function for a given variable. In such situations JaCoP uses a heuristic to find out this but it is also possible to define this explicitly. The following examples explains the process of computing derivatives of constrains.

Set<FloatVar> vars = new HashSet<FloatVar>(); 
vars.add(x1); 
vars.add(x2); 
Derivative.init(store); 
// Derivative.defineConstraint(x1x1, c0); 
// Derivative.defineConstraint(x2x2, c1); 
// Derivative.defineConstraint(x1x1x1x1, c2); 
// Derivative.defineConstraint(x2x2x2x2, c3); 
 
// ================== df/dx1 ================= 
FloatVar fx1 = Derivative.getDerivative(store, f, vars, x1); 
 
// ================== df/dx2 ================= 
FloatVar fx2 = Derivative.getDerivative(store, f, vars, x2);

In the above example, JaCoP will generate two partial derivatives -∂f
∂x1 and ∂f-
∂x2. Set vars is used to keep all variables of function defined by variable f. The function defined by variable f is usually defined by several constraints using temporary variables, such as x1x1, x2x2, x1x1x1x1 and x2x2x2x2. In such situations JaCoP tries, using heuristics, to find out what constraint defines a given temporary variable. Sometimes it is not possible and the programmer needs to define it, as indicated by the commented code and methods Derivative.defineConstraint. For example, constraint c0 defines variable x1x1. Method getDerivative of class Derivative generates a variable that defines the result of the derivative.

Derivatives, can be used for setting additional constraints on solutions. For example, when looking for minimum or maximum of a function or in the interval Newton method.

5.5.2 Multivariate Interval Newton Method

Interval Newton methods combine classical Newton method with interval analysis [12] and are used to provide better bounds on variables.

Traditionally, univariate Newton method computes a solution for equation f(x) = 0 using the following approximation.

xn+1 = xn - f(xn)-
            f′(xn)

This can be extended by using vector x and a Jacobian matrix instead of f(x). Furthermore one can extend it to interval arithmetic. This is implemented in JaCoP using “constraint” EquationSystem. This constraint takes as parameters a vector of functions and variables of these functions. The constraints computes then all partial derivatives and form a set of equations that are later solved, if possible, by interval Gauss-Seidel method.

This can be formalized as system of linear equations that, when solved, provides bounds on values (xn+1 - xn) of variables of equations.

JF (xn)(xn+1 - xn) = - F (xn)
where F : Rk Rk are continuously differentiable functions and JF(xn) is Jacobian matrix.

Below, constraint eqs specifies a set of two functions f0 and f1 of two variables x1 and x2. Constraint eqs will try to prune domains of variables x1 and x2.

EquationSystem eqs = new EquationSystem(store, 
       new FloatVar[] {f0, f1}, new FloatVar[] {x1, x2}); 
store.impose(eqs);

Chapter 6
Search

JaCoP offers methods for finding a single solution, all solutions and a solution that minimizes a given cost function. These methods are used together with methods defining variable selection and assignment of a selected value to variable. Both complete search methods and heuristic search methods can be defined in JaCoP.

JaCoP offers powerful methods for search modification, called search plug-ins. Search plug-ins can be used both for collecting information about search as well as for changing search behaviour. For more information on search plug-ins see section 6.2.

6.1 Depth First Search

A solution satisfying all constraints can be found using a depth first search algorithm. This algorithm searches for a possible solution by organizing the search space as a search tree. In every node of this tree a value is assigned to a domain variable and a decision whether the node will be extended or the search will be cut in this node is made. The search is cut if the assignment to the selected domain variable does not fulfill all constraints. Since assignment of a value to a variable triggers the constraint propagation and possible adjustment of the domain variable representing the cost function, the decision can easily be made to continue or to cut the search at this node of the search tree.

Typical search method for a single solution, for a list of variables, is specified as follows.

   Search<T> label = new DepthFirstSearch<T>();   SelectChoicePoint<T> select = new SimpleSelect<T>(var,                                               varSelect,                                               tieBreakerVarSelect                                               indomain);   boolean result = label.labeling(store, select);

where T is type of variables we are using for this search (usually IntVar or SetVar), var is a list of variables, varSelect is a comparator method for selecting variable and tieBreakerV arSelect is a tie breaking comparator method. The tie breaking method is used when the varSelect method cannot decide ordering of two variables. Finally, indomain method is used to select a value that will be assigned to a selected variable. Different variable selection and indomain methods are specified in appendix B. This search, for variables of type IntVar creates choice points xi = val and xival where xi is variable identified by variable selection comparators and val is the value determined by indomain method. For variables of type SetVar the choice is made between val xi or val∈∕xi.

The standard method can be further modified to create search for all solutions. This is achieved by adopting the standard solution listener as specified below.

   label.getSolutionListener().searchAll(true);   label.getSolutionListener().recordSolutions(true);

In the first line the flag that changes search to find all solutions is set. It is set in the default solution listener. In this example, we also set a flag that informs search to record all found solutions. If this flag is not set the search will only count solutions without storing them. The values for found solutions can be printed using label.getSolutionListener().printAllSolutions() method or the following piece of code.

   for (int i=1; i<=label.getSolutionListener().solutionsNo(); i++){ 
      System.out.print("Solution " + i + ": "); 
      for (int j=0; j<label.getSolution(i).length; j++) 
         System.out.print(label.getSolution(i)[j]); 
      System.out.println(); 
   }

Even if the solutions are not recorded, they are counted and number of found solutions can be retrieved using method label.getSolutionListener().solutionsNo().

The minimization in JaCoP is achieved by defining variable for cost and using branch-and-bound (B&B) search, as specified below.

   IntVar cost; 
   ... 
   boolean result = label.labeling(store, select, cost);

B&B search uses depth-first-search to find a solution. Each time a solution with cost costV aluei is found a constraint cost < costV aluei is imposed. Therefore the search finds solutions with lower cost until it eventually fails to find any solution that proves that the last found solution is optimal, i.e., there is no better solution.

Sometimes we want to interrupt search and report the best solution found in a given time. For this purpose, the search time-out functionality can be used. For example, 10s time-out can be set with the following statement.

   label.setTimeOut(10);

Moreover, one can define own time-out listener to perform specific actions.

6.1.1 Restart search

In some situation classical B&B algorithm is not best suited for optimization and so called restart search is used. This optimization search method finds a solution and then start search from the beginning but with additional constraint restricting the cost variable in the same way as B&B search. JaCoP does not support directly this kind of search but it can be easily implemented using the following code (use to maximize cost defined by variable cost).

   label.setSolutionListener(new CostListener<IntVar>());   store.setLevel(store.level+1);   boolean Result = true, optimalResult = false;   while (Result) {      Result = label.labeling(store, select);      store.impose(new XgtC(cost, CostValue));      optimalResult = optimalResult || Result;   }   store.removeLevel(store.level);   store.setLevel(store.level-1);

The search iteratively calls depth-first-search until no better solution is found. It also rises the store level before search and returns to “fresh” store to make it possible to operate on it later. This code requires access to value CostV alue that can be retrieved by providing a customized version of solution listener and its method executeAfterSolution. This method simply stores the value of the cost variable when a solution is found. See the code below for details.

   public class CostListener<T extends Var> extends 
                                         SimpleSolutionListener<T> { 
 
      public boolean executeAfterSolution(Search<T> search, 
                                      SelectChoicePoint<T> select) { 
         boolean returnCode = super.executeAfterSolution(search, 
                                                         select); 
 
         CostValue = cost.value(); 
         return returnCode; 
      } 
   }

6.2 Search plug-ins

The search-plugin is an object, which is informed about the current state of the search and may influence the behavior of the search. They are divided into search-plugins that change the search behavior and plugins used for collecting and sharing information. Table 6.1 lists the search-plugins available in JaCoP and their membership in a respective group.



Table 6.1: Search plug-ins available in JaCoP.


changing search cannot change search
(information sharing)


solution listener exit listener
exit child listener time-out listener
consistency listener initialize listener



The search plug-ins are called during search when they reach a specific state, as specified below.

Changing search plug-ins can override the status of the search by returning true or false status. For example, exit child listener method leftChild can override the status of the search by returning true or false status. If it returns true then the search continues and enters the right child to keep looking for a solution. Returning false instructs the search to skip exploring the right subtree

JaCoP makes it possible to combine several plugins hierarchically. Each listener may have multiple children listeners attached to it, which have potential to influence the behaviour of the parent. A very simple example of using this behavior, is using one listener to remember solution and another one to print it. These two different functionalities may be provided by two different listeners. In general, if search calls several children listeners the parent listener decides how to treat the results returned by them. The listeners already implemented in JaCoP use the following default rule to combine the return codes from different listeners. They combine their own return code with the return code of a child listener using conjunction of return codes. Several child listeners combine their return codes using disjunction of return codes.

6.3 Credit search

Credit search combines credit based exhaustive search at the beginning of the tree with local search in the rest of the tree [3]. In JaCoP, the credit search is controlled by three parameters: number of credits, number of backtracks during local search and maximum depth of search. In Figure 6.1 there is an example of the credit search tree. The search has initially 8 credits. The number of possible backtracks is three. During search half of the credits is distributed to the selected choice. The rest of the credits is distributed using the same principle for the next choice point. The first part of the search is based on the credits and makes it possible to investigate many possible assignments to domain variables while the other part is supposed to lead to a solution and can use a number of backtracks specified for this search. Moreover, the maximal depth of the search cannot be exceeded. Since we control the search it is possible to partially explore the whole tree and avoid situations when the search is stuck at one part of the tree which is a common problem of B&B algorithm when a depth first search strategy is used.


PIC

Figure 6.1: Credit search example.


An example of the command which produces the search tree depicted in Fig. 6.1 is as follows.

   SelectChoicePoint<IntVar> select = new SimpleSelect<IntVar>(vars, 
                                       new SmallestDomain<IntVar>(), 
                                       new IndomainMin<IntVar>()); 
 
   int credits=8, backtracks=3, maxDepth=1000; 
   CreditCalculator<IntVar> credit = new CreditCalculator<IntVar>( 
                                                  credits, 
                                                  backtracks, 
                                                  maxDepth); 
   Search<IntVar> search = new DepthFirstSearch<IntVar>(); 
   search.setConsistencyListener(credit); 
   search.setExitChildListener(credit); 
   search.setTimeOutListener(credit); 
 
   boolean result = search.labeling(store, select);

6.4 Limited discrepancy search

Limited discrepancy search (LDS) uses the partial search method proposed in [10]. It basically allows only a number of different decisions along a search path, called discrepancies. If the number of discrepancies is exhausted backtracking is initiated. The number of discrepancies is specified as a parameter for LDS.

An example of LDS with one discrepancy is as follows.

   Search<IntVar> label = new DepthFirstSearch<IntVar>();   SelectChoicePoint<IntVar> select = new SimpleSelect<IntVar>(var,                                      new SmallestDomain<IntVar>(),                                      new IndomainMiddle<IntVar>());   LDS<IntVar> lds = new LDS<IntVar>(2);   label.getExitChildListener().setChildrenListeners(lds);   boolean result = label.labeling(store, select);

6.5 Combining search

JaCoP offers, through its plug-ins, possibility to combine several search methods into a single complex search. For example, the following code presents a search that is build as consecutive invocation of two search methods.

   Search<IntVar> slave = new DepthFirstSearch<IntVar>(); 
   SelectChoicePoint<IntVar> selectSlave = 
          new SimpleSelect<IntVar>(vars2, 
                                   new SmallestMin<IntVar>(), 
                                   new SmallestDomain<IntVar>(), 
                                   new IndomainMin<IntVar>()); 
   slave.setSelectChoicePoint(selectSlave); 
 
   Search<IntVar> master = new DepthFirstSearch<IntVar>(); 
   SelectChoicePoint<IntVar> master = 
          new SimpleSelect<IntVar>(vars1, 
                                   new SmallestMin<IntVar>(), 
                                   new SmallestDomain<IntVar>(), 
                                   new IndomainMin<IntVar>()); 
   master.addChildSearch(slave); 
 
   boolean result = master.labeling(store, selectMaster);

When several search methods is combined to form a sequential search and the minimization is done one has to set for each sub-search a flag informing this search that it is a part of minimization. It is achieved by the following instruction.

   slave.setOptimize(true);

Chapter 7
SAT solver

JaCoP has a SAT solver that can be used together with JaCoP solver. It has been developed by Simon Cruanes and Radosław Szymanek. The SAT solver can be integrated in JaCoP as a SAT constraint and used with JaCoP search methods. This functionality is offered by a org.jacop.satwrapper.SatWrapper class. Typical translation of Boolean constraints, used in flatzinc compilations are defined in
org.jacop.satwrapper.SatTranslation.

7.1 Boolean expressions translations

Typical Boolean formulas can be added to SAT solver using class SatTranslation. This class offers translation of typical operations as and, or, not, xor, implication, all equalities and inequalities as well as reified versions of these. The following code illustrates use of the translation.

   SatTranslation sat = new SatTranslation(store); 
   sat.impose(); // impose SAT-solver as constraint 
 
   BooleanVar a = new BooleanVar(store, "a"); 
   BooleanVar b = new BooleanVar(store, "b"); 
   BooleanVar c = new BooleanVar(store, "c"); 
 
   // adding SAT clauses 
   sat.generate_implication(a, b); 
   sat.generate_and(new BooleanVar[] {a, b}, c);

The code generates Boolean clause of the form (a b) (ab = c) that is translated to CNF Boolean clause supported by the SAT solver.

(a b) 

(a b c) 

(a c) 

(b c)

The SAT translation performs several tasks. First, it defines SAT literals and binds Boolean variables to these literals. In our example, Boolean variables a, b and c are translated to literals, denoted by the same letter in the above formulas but they are different representations. Second, it adds CNF clauses representing Boolean clauses to the SAT solver. The SAT wrapper will then propagate changes between CP and SAT solver based on the evaluations of both solvers.

The search for solutions is identical with CP framework. In case of the example, we can make search on variables a, b and c. All solutions to the constraint are {0, 0, 0}, {0, 1, 0}, {1, 1, 1}

7.2 FDV encoding and constraint translations

SatWrapper offers a general way to define literals and clauses. The process is divided into two parts. First, each FDV and a inequality relation on it is bound to a literal. Then a clause is defined on the literals. We illustrate this process below.

Assume that we want to define a constraint (a 10) (b = 3 c > 7) with IntVar a, b and c. The literals for this constraint are defined as a 10, b = 3and c 7(negation of this literal is a > 7). This is encoded in JaCoP as follows.

   IntVar a = new IntVar(store, "a", 0, 100); 
   IntVar b = new IntVar(store, "b", 0, 100); 
   IntVar c = new IntVar(store, "c", 0, 100); 
 
   SatWrapper wrapper = new SatWrapper(); 
   store.impose(wrapper); 
 
   wrapper.register(a); 
   wrapper.register(b); 
   wrapper.register(c); 
 
   int aLiteral = wrapper.cpVarToBoolVar(a, 10, false); 
   int bLiteral = wrapper.cpVarToBoolVar(b, 3, true); 
   int cLiteral = wrapper.cpVarToBoolVar(c, 7, false);

In the above method cpVarToBoolVar the first parameter is the FDV variable, the second is the rhs value and the third parameter defines type if relation (false for and true for =).

When literals are defined we can generate clauses using CNF format, as presented below. First, we translate the clause to CNF form as (a 10b = 3) (a 10c 7). Then we encode this into JaCoP SAT solver.

   IntVec clause = new IntVec(wrapper.pool); 
 
   // first clause 
   clause.add(- aLiteral); 
   clause.add(bLiteral); 
   wrapper.addModelClause(clause.toArray()); 
 
   clause = new IntVec(wrapper.pool); 
 
   // second clause 
   clause.add(- aLiteral); 
   clause.add(- cLiteral); 
   wrapper.addModelClause(clause.toArray());

In the above code minus in front of the literal in statement clause.add means that the negated literal is added to the clause.

A possible solution generated by JaCoP is {0, 3, 8 }.

Chapter 8
Flatzinc

JaCoP has a front-end to flatzinc [2]. It can parse flatzinc files and execute them. Flatzinc files are usually are generated from minizinc [22] models and are “flattened” models that specify all variables, constraints and search defined in minizinc models. To fully use power of JaCoP compilation of minizinc files should used JaCoP library definitions that define translation of minizinc global constraints into JaCoP global constraints.

8.1 Minizinc Global Constraints

The minizinc to flatzinc compiler (available at http://www.minizinc.org/software.html) produces flatzinc files that can include global constraints provided by a certain solver. JaCoP has definitions of its global constraints available for minizinc in directory src/main/minizinc/org/jacop/minizinc/. This directory needs to be used by the minizinc compiler. You can make it available for the compiler in two different ways.

8.2 Minizinc Compilation and Execution

In this section we describe the compilation and execution of minizinc files using command line interface and JaCoP. JaCoP can also be added to minizinc IDE provided by NICTA http://www.minizinc.org/ide/index.html.

The typical compilation of a minizinc model is carried out by the mzn2fzn command.

mzn2fzn -G jacop model.mzn

If the model requires data files specified in dzn files the command is as follows.

mzn2fzn -G jacop model.mzn -d data.dzn

The mzn2fzn command generate two files. The file with extension fzn contains “flattened” model and file with extension ozn contains specifications on the output formatting. JaCoP usues only fzn file for solving and printing the output in the standard format. This output can be parsed by solns2out program (provided in the minizinc distribution) and printed using the format specified in minizinc model. This is automatically used in MiniZincIDE but can also be achived from the command line by using shell pipe (“|”). Typical execution of this model is provided by JaCoP in class Fz2jacop and can be done using command line skeleton specified below.

java -cp path_to_JaCoP org.jacop.fz.Fz2jacop [options] model.fzn

The execution with minizinc formating, using fzn-jacop command defined at the end of this section, is done as follows.

fzn-jacop model.fzn | solns2out model.ozn

Currently the following options are supported (printed using -h option).

Usage: java org.jacop.fz.Fz2jacop [<options>] <file>.fzn 
Options: 
    -h, --help 
        Print @this@ message. 
    -a, --all, --all-solutions 
    -v, --verbose 
    -t <value>, --time-out <value> 
        <value> - time @in@ second. 
    -s, --statistics 
    -n <value>, --num-solutions <value> 
        <value> - limit on solution number. 
    -b, --bound - use bounds consistency whenever possible; 
        override annotation ":: domain" and select constraints 
        implementing bounds consistency (@default false@). 
    -sat use SAT solver @for boolean@ constraints. 
    -cs, --complementary-search - @try@ to gather all model, non-introduced 
         variables to define the @final or default@ search, instead @of using@ 
         @output@ variables only. 
    -i, --interval print intervals instead @of values for@ floating variables 
    -p <value>, --precision <value> defines precision @for@ floating operations 
        overrides precision definition @in@ search annotation.

It is practical to create a script (called for example fzn-jacop) that can execute JaCoP on flatzinc file. Below is a simple bash script (assuming jacop.jar is in the current directory), for example.

#!/bin/bash 
exec java -cp jacop.jar org.jacop.fz.Fz2jacop "$@"

8.3 Flatzinc Extensions

JaCoP offers possibility to extend flatzinc models execution with JaCoP specific methods. They are specified in this section.

8.3.1 Flatzinc Loader

In some situations it is useful to parse the model, create all variables and constraints and then use JaCoP to handle the search and possibly other processing. This functionality is provided by class FlatzincLoader. This class implements methods for loading a flatzinc model and methods for getting information about the model.

An example illustrating how to use this functionality to built a flatzinc solver is implemented in org.jacop.examples.flatzinc.FlatzincSolver.

8.3.2 Floating-Point Minimization

Standard minimization method is based on branch-and-bound method. This means that whenever a solution is found the next solution must have its cost lower than the cost of the current solution. This method work fine for IntVar cost functions but it does not work efficiently for FloatVar. Therefore JaCoP offers another method, called Optimization, described in section 5.3. This method tries first to find a solution for the model. If such solution exists it tries to narrow the interval for the cost variable to left half and searches for a solution. If such solution exists it goes again to the left sub-interval. If there is no solution it tries the right sub-interval. The process stops when the cost variable interval is considered as ground (within a given precision). This method is implemented, for flatzinc models, in org.jacop.examples.flatzinc.FloatMinimize.

Chapter 9
Scala Wrapper

JaCoP can also be used from Scala [1]. Scala wrapper (org.jacop.scala) is defined to overload basic operations, equalities and inequalities, define global constraints as well as make search specifications simpler.

9.1 Example

The following example presents a classical CP example SendMoreMoney written in Scala and using JaCoP library.

The example imports JaCoP wrapper in line 1 and use trait jacop. In this example, in lines 5-12 we define finite domain variables used in the model as well as a vector of these variables in line 14. Alldifferent constraint is defined in line 16. Then arithmetic equality between two sums is define in lines 18-19. Finally, constraint that s and m must be greater than zero are defined. Notice, that Scala wrapper overloads all arithmetical operations, such as + in the example, and inequalities and equalities (they are prefixed with #).

Finally, the search is defined for the model. It is build of the method that requests search for a single solution satisfying the model constraints and search specification (method search).

For more details, please check the next subsections of this chapter.

1import org.jacop.scala._ 
2 
3object SendMoreMoney extends App with jacop { 
4 
5  val s = new IntVar("s", 0, 9) 
6  val e = new IntVar("e", 0, 9) 
7  val n = new IntVar("n", 0, 9) 
8  val d = new IntVar("d", 0, 9) 
9  val m = new IntVar("m", 0, 9) 
10  val o = new IntVar("o", 0, 9) 
11  val r = new IntVar("r", 0, 9) 
12  val y = new IntVar("y", 0, 9) 
13 
14  val fd = Array(s,e,n,d,m,o,r,y) 
15 
16  alldifferent( fd ) 
17 
18  1000*s + 100*e + 10*n + d + 1000*m + 100*o + 10*r + e #= 
19       10000*m + 1000*o + 100*n + 10*e + y 
20 
21  s #> 0 
22  m #> 0 
23 
24  val result = satisfy( search(fd, input_order, indomain_min), 
25                        printSol ) 
26 
27  statistics 
28 
29  def printSol() : Unit = { 
30    print("Solution: ") 
31      for (v <- fd) 
32        print( v+" " ) 
33      println 
34  } 
35}

Our model uses multiplications and additions to form a weighted sum of variables in lines 18-19. This is decomposed to JaCoP constraints XmulYeqZ, XplusYeqZ and XeqY that results in 39 variables and 25 constraints. The solution found by this program is presented below.

Solution: s = 9 e = 5 n = 6 d = 7 m = 1 o = 0 r = 8 y = 2 
 
Search statistics: 
================== 
Search nodes : 7 
Propagations : 504 
Search decisions : 4 
Wrong search decisions : 3 
Search backtracks : 0 
Max search depth : 4 
Number solutions : 1

The program can be further improved by using the global constraint intended for such situations. To use JaCoP LinearInt constraint one can make the following specification. It resuts in 10 variables and 6 constraints.

sum( Array(s, e, n, d, m, o, r, e), 
     Array(1000, 100, 10, 1, 1000, 100, 10, 1) ) #= 
  sum( Array(m,o, n, e, y), Array(10000, 1000, 100, 10, 1) )

or to use a single LinearInt constraint we can use the following code. It has 9 variables and 5 constraints.

  sum( Array(s, e, n, d, m, o, r, e, m,o, n, e, y), 
       Array(1000, 100, 10, 1, 1000, 100, 10, 1, 
             -10000, -1000, -100, -10, -1) ) #= 0

The solution to the last model is the same as before but uses only 3 search nodes and has 21 propagations. This is caused by more efficient linear constraint and its presolving (eliminating multiple variables in the constraint by summing up their weights).

9.2 Scala Model

All variables, constraints and other solver specific data are kept in the store, defined by class Store in JaCoP. Scala wrapper extends Store and defines class Model. Model is created implicitly. If needed one can access it using method getModel and set a new Model using method setModel(new Model()).

9.3 Variables

In Scala wrapper one can define four types of variables: IntVar, BoolVar, SetVar and FloatVar as well as constraints over these variables. Since the store is defined implicitly all variables can be created by specifying their names and domains only (e.g., lines 5-12 in the SendMoreMoney example). BoolVar do not have specification of domain since it is given as domain 0/1.

For convenience, a set of integers is defined as IntSet and can be used to build domains for variables. This class has operations, such as union (+), intersection (*), subtraction (\) and set complement (~) defined. Therefore, domains can be defined as illustrated on the examples below.

  // definition of x1::{1..10, 13} 
  val set1 = new IntSet(1,10) + 13 
  val x1 = new IntVar("x1", set1) 
 
  // definition of x2::{1..10, 13..15} 
  val set2 = new IntSet(1,10) + new IntSet(13, 15) 
  val x2 = new IntVar("x2", set2)

9.4 Constraints

One can use all JaCoP constraints, defined in Java, directly in a Scala program but for convenience some methods are overloaded to provide more intuitive way of specifying constraints.

For IntVar operators, +, -, *, div, mod and   are defined. Moreover, the following equality and inequality are defined #=, #\=, #<, #<=, , #>, #>=. Constraint that requires a variable value to be in a set is defined as in operator.

Similar operators are defined for FloatVar. In this case, however, mod and in are not applicable. Additionally, there constraints defining mathematical functions ad defined in Table 9.1





Function Example JaCoP constraint



i=0Nxi v = sum(x_vector) LinearFloat
i=0Nai xi v = sum(x_vector, a_vector) LinearFloat
v = |x| v = abs(x) AbsPeqR
v = ex v = exp(x) ExpPeqR
v = ln(x) v = ln(x) LnPeqR
v = √--
 x v = sqrt(x) SqrtPeqR
v = sin(x) v = sin(x) SinPeqR
v = asin(x) v = asin(x) AsinPeqR
v = cos(x) v = cos(x) CosPeqR
v = acos(x) v = acos(x) AcosPeqR
v = tan(x) v = tan(x) TanPeqR
v = atan(x) v = atan(x) AtanPeqR




Table 9.1: Floating point constraints.


Set variables have the following operators: + (union), * (intersection), \ (set subtraction), <> (disjoint sets), in (set inclusion) and method card (set cardinality). Set inequalities are the following: #= (equality), #<= (lexicographically less or equal) #>= (lexicographically greater or equal).

Boolean variables have the following operators: #= (equality), /\ (and), \/ (or), xor (exclusive or), ~ (negation), -> (implication) and <=> (reification). Implication and reification take a Boolean variable and a primitive constraint as parameters. In reification an order of parameters does not matter.

In the object package of the JaCoP wrapper there are defined global constraints and therefore they can be directly used in programs. For example, in the SendMoreMoney program we use alldifferent constraint in line 16. Table 9.2 presents a list of global constraints avaible through the wrapper.





Name Example JaCoP constraint



alldifferent alldifferent(vector) Alldifferen
alldistinct alldistinct(vector) Alldistinct
gcc gcc(vector1, vector2) GCC
sum sum(vector, result) SumInt
result = sum(vector) or SumBool
result = sum(vector, intVector) LinearInt
sumDom result = sumDom(vector, intVector) LinearIntDom
abs result = abs(v) AbsXeqY
min result = min(v) Min
min(vector, result) Min
max result = max(vector) Max
max(vector, result) Max
count result = count(vector, value) Count
count(vector, result, value) Count
values result = values(vector) Values
values(vector, result) Values
element element(index, vector, value) ElementInteger
value = vectore(index) or ElementVariable
diff2 diff2(rectnagles) Diff2
diff2(x_vect, y_vect, lx_vect, ly_vect)
cumulative cumulative(t_vect, d_vect, r_vect, limit) Cumulative
circuit circuit(vector) Circuit
subcircuit subcircuit(vector) Subcircuit
assignment assignment(vector1, vector2) Assignment
among among(vector, set, result) Among
among(vector1, vector2, result) AmongVar
knapsack knapsack(profits, weights, quantity) Knapsack
regular regular(dfaa , vector) Regular
clause clause(vectorP, vectorN) Clause
arg_min result = arg_min(vector) ArgMin
arg_max result = arg_max(vector) ArgMax
sequence sequence(vector. set, q, min, max) Sequence
stretch stretch(vector, min_vect, max_vect, x_vect) Stretch
lex lex(vector1, vector2) LexOrder
lex(matrix) Lex
softAlldifferent softAlldifferent(vector, cost) SoftAlldifferent
softGCC softGCC(vector, hardLB, hardUB, softGCC
    values, softCount))
AND AND(constraints_vect) And
OR OR(constraints_vect) Or
NOT NOT(constraint) Not




Table 9.2: Global constraints available from Scala wrapper.


9.5 Search

Scala wrapper offers several search methods as well as possibility to select methods for variable selection and value selection. In the example in this section, search is defined to find a solution that satisfies constraints. Method search defines input_order variable selection method and indomain_min value selection method. Moreover, method satisfy passes method printSol that is used to print a solution when it is found.

The search methods defined in Scala wrapper are presented in Table 9.3.




Name Desctiption


satisfy Find a single solution for vector of variables
satisfy_seq Find a single solution for sequence of searches
satisfyAll Find all solutions for vector of variables
satisfyAll_seq Find all solutions for sequence of searches
minimize Find a solution with minimal cost for vector of variables
maximize Find a solution with maximal cost for vector of variables
minimize_seq Find a solution with minimal cost for sequence of searches
maximize_seq Find a solution with amximal cost for sequence of searches



Table 9.3: Search methods.





Name Description


search Defines vector of variables, variable and value selection methods for IntVar
search_vector Defines vector of vectors of variables, variable and value selection methods
search_splita Defines vector of variables and variable selection methods for split search
search_floatb Defines vector of variables, variable and value selection methods for FloatVar


Table 9.4: Selection methods for search.




Name JaCoP method


input_order null
first_fail SmallestDomain
anti_first_fail LargestDomain
most_constrained MostConstrainedStatic
smallest_min SmallestMin
smallest SmallestMin
smallest_max SmallestMax
largest LargestMax
max_regret MaxRegret



Table 9.5: Variable selection heuristics.





Name JaCoP method


first_fail_set MinCardDiff
anti_first_fail_set MaxCardDiff
most_constrained_set MostConstrainedStatic
min_glb_card MinGlbCard
min_lub_card MinLubCard



Table 9.6: Variable selection heuristics for SetVar.





Name JaCoP method


indomain_min IndomainMin
indomain_maX IndomainMax
indomain_middle IndomainMiddle
indomain_median IndomainMedian
indomain_random IndomainRandom



Table 9.7: Value selection heuristics.





Name JaCoP method


indomain_min_set IndomainSetMin
indomain_max_set IndomainSetMax
indomain_random_set IndomainSetRandom



Table 9.8: Value selection heuristics for SetVar.


9.6 Other Methods

Scala wrapper implements several methods that can be used in the model. For floating-point solver, there are defined two methods for setting and checking current definition of precision in the solver.

setPrecision(1e-12)  // set floating-point solver precision to 10-12precision()          // get floating-point solver precision

Search statistics can be obtained using method

statistics

as used in the example in this chapter.

Appendix A
JaCoP constraints

A.1 Arithemtic constraints



Primitive Constraint JaCoP specification


X = Const XeqC(X, Const)
X = Y XeqY(X, Y)
XConst XneqC(X, Const)
XY XneqY(X, Y)
X > Const XgtC(X, Const)
X > Y XgtY(X, Y)
X Const XgteqC(X, Const)
X Y XgteqY(X, Y)
X < Const XltC(X, Const)
X < Y XltY(X, Y)
X Const XlteqC(X, Const)
X Y XlteqY(X, Y)
X Const = Z XmulCeqZ(X, Const, Z)
X + Const = Z XplusCeqZ(X, Const, Z)
X + Y = Z XplusYeqZ(X, Y, Z)
X + Y + Const = Z XplusYplusCeqZ(X, Y, Const, Z)
X + Y + Q = Z XplusYplusQeqZ(X, Y, Q, Z)
X + Const Z XplusClteqZ(X, Const, Z)
X + Y Z XplusYlteqZ(X, Y, Z)
X + Y > Const XplusYgtC(X, Y, Const)
X + Y + Q > Const XplusYplusQgtC(X, Y, Q, Const)
|X| = Y AbsXeqY(X, Y)


Non-primitive Constraint JaCoP specification


X Y = Z XmulYeqZ(X, Y, Z)
X ÷ Y = Z XdivYeqZ(X, Y, Z)
X modY = Z XmodYeqZ(X, Y, Z)
XY = Z XexpYeqZ(X, Y, Z)

A.2 Set constraints



Primitive Constraint JaCoP specification


e A EinA(e, A)
S1 = S2 AeqB(S1, S2)
S1 S2 AinB(S1, S2)


Non-primitive Constraint JaCoP specification


S1 S2 = S3 AunionBeqC(S1, S2, S3)
S1 S2 = S3 AintersectBeqC(S1, S2, S3)
S1 \ S2 = S3 AdiffBeqC(S1, S2, S3)
S1 <> S2 AdisjointB(S1, S2)
Match Match(Set, VarArray)
#S = C CardA(S, C)
#S = X CardAeqX(S, X)
Weighted sum < S, W > = X SumWeightedSet(S, W, X)
Set[X] = Y ElementSet(X, Set, Y)

A.3 Floating point constraints



Primitive Constraint JaCoP specification


P = Const PeqC(P, Const)
P = Q PeqQ(P, Q)
PConst PneqC(P, Const)
PQ PneqQ(P, Q)
P > Const PgtC(P, Const)
P > Q PgtQ(P, Q)
P Const PgteqC(P, Const)
P Q PgteqQ(P, Q)
P < Const PltC(P, Const)
P < Q PltQ(P, Q)
P Const PlteqC(P, Const)
P Q PlteqQ(P, Q)
P + Const = R PplusCeqR(P, Const, R)
P + Q = R PplusQeqR(P, Q, R)
P - Const = R PminusCeqR(P, Const, R)
P - Q = R PminusQeqR(P, Q, R)


Non-primitive Constraint JaCoP specification


P Const = R PmulCeqR(P, Const, R)
P Q = R PmulQeqR(P, Q, R)
P∕Q = R PdivQeqR(P, Q, R)
P∕Const = R PdivCeqR(P, Const, R)
√P-- = R SqrtPeqR(P, Q)
sin(P) = R SinPeqR(P, Q)
cos(P) = R CosPeqR(P, Q)
tan(P) = R TanPeqR(P, Q)
asin(P) = R AsinPeqR(P, Q)
acos(P) = R AcosPeqR(P, Q)
atan(P) = R AtanPeqR(P, Q)
ln(P) = Q LnPeqQ(P, Q)
exp(P) = Q ExpPeqQ(P, Q)
|P| = Q AbsPeqQ(P, Q)
V [Index] = V alue Element(Index, V, Value)
X = P XeqQ(X, P) where x is IntVar and P is FloatVar

A.4 Logical, conditional and reified constraints



Primitive Constraint JaCoP specification


¬c Not(c);
c1 c2 ⋅⋅⋅cn PrimitiveConstraint[] c = {c1, c2, …cn};
Or(c);
or
ArrayList c =
   new ArrayList();
c.add(c1); c.add(c2); …c.add(cn);
Or(c);
c1 c2 ⋅⋅⋅cn PrimitiveConstraint[] c = {c1, c2, …cn};
And(c);
or
ArrayList c =
   new ArrayList();
c.add(c1); c.add(c2); …c.add(cn);
And(c);
a1 ... an b1 ... bm BooleanVar[] a = {a1, a2, …, an};
BooleanVar[] b = {b1, b2, …, bm};
BoolClause(a, b)
c1 c2 Eq(c1, c2);
X in Dom In(X, Dom);
c B Reified(c, B);
c ⇔¬B Xor(c, B);
if c1 then c2 IfThen(c1, c2);
if c1 then c2 else c3 IfThenElse(c1, c2, c3);




Primitive constraints JaCoP specification
on Boolean variables


BooleanVar[] b = {b1, b2, …, bn};
or
ArrayList b =
   new ArrayList
b.add(b1); b.add(b2); …b.add(bn);
BooleanVar result =
   new BooleanVar(store, "result");
result = b1 b2 ⋅⋅⋅bn AndBool(b, result)
result = b1 b2 ⋅⋅⋅bn OrBool(b, result)
result = b1 b2⋅⋅⋅bn XorBool(b, result)
result = b1 b2 IfThenBool(b1, b2, result)
result = b1 == b2 == ⋅⋅⋅ == bn EqBool(b, result)

A.5 Global constraints

SumInt, SumBool, LinearInt
used with ==, <, >, <=, >=, !=

(examples with == only)

x1 + x2 + ⋅⋅⋅ + xn = sum
Primitive constraint

IntVar[] x = {x1, x2, …, xn};
IntVar sum = new IntVar(…)
SumInt(store, x, ==, sum);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
IntVar sum = new IntVar(…)
SumInt(store, x, ==, sum);

b1 + b2 + ⋅⋅⋅ + bn = sum
Primitive constraint

BooleanVar[] b = {b1, b2, …, bn};
IntVar sum = new IntVar(…)
SumBool(store, b, ==, sum);
or
ArrayList b = new ArrayList();
b.add(b1); b.add(b2); …b.add(bn);
IntVar sum = new IntVar(…)
SumBool(store, b, ==, sum);
bi can also be 0/1 IntVar

w1 x1 + w2 x2 + ⋅⋅⋅ + wn xn = sum
Primitive constraint

IntVar[] x = {x1, x2, …, xn};
IntVar sum = new IntVar(…)
int[] w = {w1, w2, …, wn};
LinearInt(store, x, w, ==, sum);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
IntVar sum = new IntVar(…)
ArrayList w=new ArrayList();
w.add(w1); w.add(w1); …w.add(wn);
LinearInt(store, x, w, ==, sum);

LinearIntDom(store, x, w, ==, sum); domain consistency (== and !=)

alldifferent([x1,x2,,xn])

IntVar[] x = {x1, x2, …, xn};
Alldifferent(x);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
Alldifferent(x);

alldiff([x1,x2,,xn])

IntVar[] x = {x1, x2, …, xn};
Alldiff(x);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
Alldiff(x);

alldistinct([x1,x2,,xn])

IntVar[] x = {x1, x2, …, xn};
Alldistinct(x);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
Alldistinct(x);

among([x1,x2,,xn],val,count)

IntVar[] x = {x1, x2, …, xn};
IntervalDomain val = new IntervalDomain(k,l);
IntVar count = new IntVar(…);
Among(x, val, count);

amongVar([x1,x2,,xn],[y1,y2,,ym],count)

IntVar[] x = {x1, x2, …, xn};
IntVar[] y = {y1, y2, …, ym};
IntVar count = new IntVar(…);
Among(x, y, count);

assignment([x1,x2,,xn],[y1,y2,,yn])

IntVar[] x = {x1, x2, …, xn};
IntVar[] y = {y1, y2, …, yn};
Assignment(x, y);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
ArrayList y = new ArrayList();
y.add(y1); y.add(y2); …y.add(yn);
Assignment(x, y);

circuit([x1,x2,,xn])

IntVar[] x = {x1, x2, …, xn};
Circuit(x);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
Circuit(x);

subcircuit([x1,x2,,xn])

IntVar[] x = {x1, x2, …, xn};
Subcircuit(x);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
Subcircuit(x);

count(value,[x1,x2,,xn],var)

int value = …;
IntVar var = new IntVar(…);
IntVar[] x = {x1, x2, …, xn};
Count(x, var, value);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
Count(x, var, value);

cumulative([t1,t2,,tn],[d1,d2,,dn],[r1,r2,,rn],ResourceLimit)

IntVar[] t ={t1, t2, …, tn};
IntVar[] d ={d1, d2, …, dn};
IntVar[] r = {r1, r2, …, rn};
IntVar Limit = new IntVar(…);
Cumulative(t, d, r, Limit);2
or using ArrayList

diff2([[x1,y1,dx1,dy1],,[xn,yn,dxn,dyn]])

IntVar[][] r = {{x1,y1,dx1,dy1}, …,
{xn,yn,dxn,dyn}};
Diff(r); or Diff2(Store, r);1
or using ArrayList>

or

diff2([x1,,xn],[y1,,yn],[dx1,,dxn],[dy1,,dyn])

IntVar[] x = {x1, …, xn};
IntVar[] y = {y1, …, yn};
IntVar[] dx = {dx1, …, dxn};
IntVar[] dy = {dy1, …, dyn};
Diff(x, y, dx, dy); or Diff2(Store, x, y, dx, dy);1
or using ArrayList

distance(x,y,dist)

IntVar x, y, dist;
Distance(x, y, dist);

element(Index,[n1,n2,,nn],V alue)

IntVar Index, Value;
int[] i = {n1, n2, …, nn };
Element(Index, i, Value);

element(Index,[x1,x2,,xn],V alue)

IntVar Index, Value;
IntVar[] x = {x1, x2, …, xn };
Element(Index, x, Value);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
Element(Index, x, Value);

extensionalSupport([x1,x2,,xn],{{1,2,,n},{},,{}})
extensionalConflict([x1,x2,,xn],{{1,2,,n},{},,{}})

IntVar[] x = {x1, x2, …, xn};
int[][] intTuple = {{…}, …});
ExtensinalSupportVA(x,intTuple);
or
ExtensinalConflictVA(x,intTuple);
or
ExtensinalSupportSTR(x,intTuple);
or
ExtensinalSupportMDD(x,intTuple);

gcc([x1,x2,,xn],[y1,y2,,ym])

IntVar[] x = {x1, x2, …, xn};
IntVar[] y = {y1, y2, …, ym};
GCC(x, y);

min([x1,x2,,xn],Xmin)

IntVar[] x = {x1, x2, …, xn};
Min(x, Xmin);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
Min(x, Xmin);

max([x1,x2,,xn],Xmax)

IntVar[] x = {x1, x2, …, xn};
Max(x, Xmax);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
Max(x, Xmax);

arg_min([x1,x2,,xn],indexMin)

IntVar[] x = {x1, x2, …, xn};
ArgMin(x, indexMin);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
ArgMin(x, indexMin);

arg_max([x1,x2,,xn],indexMax)

IntVar[] x = {x1, x2, …, xn};
Max(x, indexMax);
or
ArrayList x = new ArrayList();
x.add(x1); x.add(x2); …x.add(xn);
Max(x, indexMax);

knapsack(profits,weights,quantity,knapsackCapacity,knapsackProfit)

int[] profits = {p1, p2, …, pn};
int[] weights = {w1, w2, …, wn};
IntVar[] quantity = {q1, q2, …, qn};
IntVar knapsackCapacity = new IntVar(…);
IntVar knapsackProfit = new IntVar(…);
Knapsack(profits, weights, quantity, knapsackCapacity, knapsackProfit);

geost(objects,constraints,shapes)

IntVar xOrigin = new IntVar(store, "x1", 0, 20);
IntVar yOrigin = new IntVar(store, "y1", 0, 5);
IntVar shapeNo = new IntVar(store, "s1", 1, 1);
IntVar startGeost = new IntVar(store, "start"+1, 0, 0);
IntVar durationGeost = new IntVar(store, "duration"+1, 1, 1);
IntVar endGeost = new IntVar(store, "end"+1, 1, 1);
IntVar[] coords = {xOrigin, yOrigin};
int objectId = 1;
GeostObject o = new GeostObject(objectId, coords, shapeNo, startGeost, durationGeost, endGeost);
ArrayList objects = new ArrayList();
objects.add(o);
int[] origin = {0, 0};
int[] length = {10, 2};
Shape shape = new Shape(j, new DBox(origin, length));
ArrayList shapes = new ArrayList();
shapes.add(shape);
int[] dimensions = {0, 1};
NonOverlapping constraint = new NonOverlapping(objects, dimensions);
ArrayList constraints = new ArrayList();
constraints.add(constraint);
store.impose(new Geost(objects, constraints, shapes));

regular(fsm,[x1,x2,,xn])

FSM fsm = new FSM();
IntVar[] x = {x1, x2, …, xn};
Regular(fsm, x);

sequence([x1,x2,,xn],set,q,min,max)

IntVar[] x = {x0, x1 …xn};
IntervalDomain set = new IntervalDomain(…);
int q, main, max;
Sequence(x, set, q, min, max);

stretch(values,min,max,[x1,x2,,xn])

int[] values, main, max;
IntVar[] x = {x0, x1 …, xn};
Stretch(values, min, max, x);

values([x1,x2,,xn],count)

IntVar[] x = {x0, x1 …, xn};
IntVar count = new IntVar(…);
Values(x, count);

lex([[x11,x12,,x1n],,[xk1,xk2,,xkm]])

IntVar[][] x = {{x0, x1 …, xn}, …};
Lex(x);
or
Lex(x, true);

lex([x1,x2,,xn],[y1,y2,,ym])

IntVar[] x = {{x0, x1 …, xn};
IntVar[] y = {{y0, y1 …, yn};
LexOrder(x, y);
or
LexOrder(x, y, true);

soft-alldifferent([x1,x2,,xn],cost,violation_measure)

IntVar[] x = {x0, x1 …, xn};
IntVar count = new IntVar(…);
SoftAlldifferent(x, cost, ViolationMeasure.DECOMPOSITION_BASED);
or
SoftAlldifferent(x, cost, ViolationMeasure.VARIABLE_BASED);

soft-GCC([x1,x2,,xn],hardCounters,countedV alues,softCounters,cost,violation_measure)

IntVar[] x = {x0, x1 …, xn};
IntVar[] hardCounters = {h1, h2, …, hn};
int[] countedValues = {v1, v2, …, vn};
IntVar[] softCounters = {s1, s2, …, sn};
SoftGCC(x, hardCounters, countedValues, softCounters, cost, ViolationMeasure.VALUE_BASED); or
other constructors (see API specification).

Appendix B
JaCoP search methods

B.1 Variable and value selection for IntVar

B.2 Variable and value selection for SetVar

B.3 Variable selection for FloatVar

B.4 Search methods

We specify search methods for finite domain variables (IntVar). Similar methods can be defined for set variables (SetVar).

B.5 Important methods for search plug-ins

Appendix C
JaCoP debugging facilities

C.1 Available switches

Most important debugging facility is made available through the Boolean variables of the Switch class. In order to use this you have to have specially compiled JaCoP library where all mas