Maximal Sets of Solutions for Constraint Satisfaction Problems

Real constrained problems often demand specific answers to meet requirements like bounded computation time, user preferences or data changes. In the framework of Constraint Satisfaction Problems, sets of solutions which are maximal cartesian products of sub-domains provide a large amount of information in a synthetic and exploitable fashion. In this paper, standard Forward-Checking is extended and a heuristic is designed to compute such sets via their intensional representation. From experiments and a theoretical analysis, we demonstrate the interest of computing a single set instead of individual solutions. 1 SETS OF SOLUTIONS FOR CSPs A Constraint Satisfaction Problem (CSP,[8],[9]) is defined by a set of variables and a set of constraints. Each variable ranges over a finite domain of values and each constraint specifies which values of the variables are compatible with each other. A solution of a CSP is an assignment of values to all the variables such that all the constraints are satisfied. Recent works address the task of computing the set of solutions or certain subsets. In [3], Freuder introduces the notion of interchangeability to cluster values of a variable that belong to the same solutions. Removing some interchangeable values before generating all the solutions using Forward-Checking has been proved successful on loose and sparse-connected problems (see [1]). In [6], Backtracking and Forward-Checking are extended to operate on a cross product representation of partial consistent assignments. The corresponding algorithms BT-CPR and FC-CPR generate a partition of the set of solutions whose elements are cross products of sub-domains of the variables. This approach guarantees that no additional constraint check is made when seeking all solutions or proving the inconsistency of a CSP. A similar approach is developed in [5]. Yet the computation of all the solutions is often untractable, either because their number is prohibitive (eg loose problems of great size), their set is not easily factorable (eg n-queens problems), or simply because the problem is too difficult. In these cases, one has to search for a subset of the solutions. With regard to applications, such a set should satisfy the following basic criteria: proximity between the solutions, easiness of the computation and compactness of the representation. One may also wish a precise number of solutions. In fact, additional solutions should be computed as long as their cost of computation is negligible compared to the overall cost. The criterion of proximity can be expressed as follows: the least solutions differ from one another in terms of values, the better. Having a set of close solutions at one's disposal is indeed desirable when new constraints appear. If a solution becomes invalid, a close solution may satisfy the new problem and if so, its real implementation will cause very localized changes. For instance, one would like to preserve a scheduling in progress when a failure occurs on a machine: assigning a new machine (value) to the unallocated job (variable) should affect a minimum 1 Centre d'Etudes et de Recherches de Toulouse, Département d'Informatique, 2 avenue Belin, 31055 Toulouse, FRANCE number of existing allocations. In this perspective, the number of close solutions one desires should be as large as possible. For this purpose, we propose to compute a subset of the solutions of a problem, called a maximal consistent decision (denoted MCD, see [7]). A MCD is a cartesian product of subdomains that only contains solutions and that cannot be increased by any value on any variable to yield a greater MCD. The interest of a MCD is to provide a set of close solutions, maximal in size, and easily exploitable via its synthetic representation. This paper describes the algorithm Generalized-ForwardChecking (GFC) which extends Forward-Checking (FC) to compute all the MCDs of a general CSP. Dedicated to the generation of one MCD, GFC is compared theoretically and practically to FC. Experiments show that both algorithms behave identically on inconsistent problems and that the extrawork to get a MCD instead of a single solution is negligible for most consistent problems. The paper is organized as follows: section 2 recalls basic definitions and properties of MCDs; section 3 introduces GFC; section 4 establishes complexities of backtrack-free searchs for MCDs and solutions; section 5 presents the experiments on randomly generated CSPs; section 6 is a brief conclusion. 2 MAXIMAL CONSISTENT DECISIONS A Constraint Satisfaction Problem is defined by (X,D,C,R) where: • X={X1,...Xn} is the set of variables, • D={D1,...Dn} is the set of domains where Di is the set of the possible values for variable Xi, • C={C1 , ... C m } is the set of constraints where constraint Cj=(Xi1,Xi2,...Xik)⊆X is identified with the tuple of variables it involves, • R={R1,...Rm} is the set of relations where Rj⊆Di1×Di2...×Dik specifies the tuples of values that satisfy constraint Cj=(Xi1,Xi2,...Xik). Subsequently, n denotes the number of variables, m the number of constraints, a the maximum arity of constraints and d the maximum domain size. We first define decisions as sub-domain assignments to variables: Def in i t ion 1 A decision δ on (Xi1,Xi2,...X ik) is a k-tuple (δi1,δi2,...δik) whose j-th component δij is a non-empty subset of Dij the domain of Xij, for j=1...k. The cartesian product of the components δi1×δi2×...×δik is the set of implicit assignments of δ. If these assignments are locally consistent, δ is said to be consistent: Def in i t ion 2 A decision δ on (Xi1,Xi2,...Xik) is consistent if every implicit assignment of δ satisfies all the constraints Cj⊆(Xi1,Xi2,...Xik). For the problem of figure 1, decision ({0},{0,1},{1}) on (X1,X2,X3) is consistent. Adding value 2 of X2 provides a greater