Pedro Meseguer

Constraint Satisfaction Problems: An Overview

Many AI problems can be formulated as Constraint Satisfaction Problems CSP. Using a systematic search process with backtracking this kind of problems can be solved, but this method is very inefficient. Other methods solving CSP have been developed showing a better performance. In this paper an overview of methods algorithms and heuristics solving CSP is presented, stressing the key ideas underlying the different methods.

Asynchronous backtracking without adding links: a new member in the ABT family

Following the pioneer work of Yokoo and colleagues on the ABT (asynchronous backtracking) algorithm, several ABT-based procedures have been proposed for solving distributed constraint networks. They differ in the way they store nogoods, but they all use additional communication links between unconnected agents to detect obsolete information. In this paper, we propose a new asynchronous backtracking algorithm which does not need to add links between initially unconnected agents. To make the description simpler and to facilitate the comparisons between algorithms, we present a unifying framework from which the new algorithm we propose, as well as existing ones, are derived. We provide an experimental evaluation of these algorithms.

Maintaining reversible DAC for Max-CSP

Abstract We introduce an exact algorithm for maximizing the number of satisfied constraints in an overconstrained CSP (Max-CSP). The algorithm, which can also solve weighted CSP, probabilistic CSP and other similar problems, is based on directed arc-inconsistency counts (DAC). The usage of DAC increases the lower bound of branch and bound based algorithms for Max-CSP, improving their efficiency. Originally, DAC were defined following a static variable ordering. In this paper, we relax this condition, showing how DAC can be defined from a directed constraint graph. These new graph-based DAC can be effectively used for lower bound computation. Interestingly, any directed constraint graph of the considered problem is suitable for DAC computation, so the selected graph can change dynamically during search, aiming at optimizing the exploitation of directed arc-inconsistencies. In addition, directed arc-inconsistencies are maintained during search, propagating the effect of value pruning. With these new elements we present the PFC maintaining reversible DAC algorithm (PFC-MRDAC), a natural successor of PFC-DAC for Max-CSP. We provide experimental evidence for the superiority of PFC-MRDAC on random and real overconstrained CSP instances, including problems with weighted constraints.

Solving Max-SAT as weighted CSP

For the last ten years, a significant amount of work in the constraint community has been devoted to the improvement of complete methods for solving soft constraints networks. We wanted to see how recent progress in the weighted CSP (WCSP) field could compete with other approaches in related fields. One of these fields is propositional logic and the well-known Max-SAT problem. In this paper, we show how Max-SAT can be encoded as a weighted constraint network, either directly or using a dual encoding. We then solve Max-SAT instances using state-of-the-art algorithms for weighted Max-CSP, dedicated Max-SAT solvers and the state-of-the-art MIP solver CPLEX. The results show that, despite a limited adaptation to CNF structure, WCSP-solver based methods are competitive with existing methods and can even outperform them, especially on the hardest, most over-constrained problems. This research is partially supported by the French-Spanish collaboration PICASSO 05158SM - Integrated Action HF02-69. The second and third authors are also supported by the REPLI project TIC-2002-04470-C03.

Distributed Dynamic Backtracking

In the scope of distributed constraint reasoning, the main algorithms presented so far have a feature in common: the addition of links between previously unrelated agents, before or during search. Our work presents a new search procedure for finding a solution in a distributed constraint satisfaction problem. This algorithm makes use of some of the good properties of centralized dynamic backtracking. It is sound, complete and allows a high level of asynchronism by sidestepping the unnecessary addition of links.

Exploiting symmetries within constraint satisfaction search

Abstract Symmetry often appears in real-world constraint satisfaction problems, but strategies for exploiting it are only beginning to be developed. Here, a framework for exploiting symmetry within depth-first search is proposed, leading to two heuristics for variable selection and a domain pruning procedure. These strategies are then applied to two highly symmetric combinatorial problems, namely the Ramsey problem and the generation of balanced incomplete block designs. Experimental results show that these general-purpose strategies can compete with, and in some cases outperform, previous more ad hoc procedures.

Distributed forward checking

A reason to distribute constraint satisfaction is privacy: agents may not want to share their values, and they may wish to keep constraints as private as possible. In this paper, we present the Distributed Forward Checking algorithm, a natural successor of Asynchronous Backtracking, where some privacy is achieved on agent values. Regarding constraints, we introduce the Partially Known Constraints model, which allow a constraint between two agents to be not completely known by any of them. With these elements, we obtain new solving algorithms that enforce privacy and maintain completeness. Empirical results are provided. This research is supported by the REPLI project TIC-2002-04470-C03-03.

On forward checking for non-binary constraint satisfaction

Solving non-binary constraint satisfaction problems, a crucial challenge today, can be tackled in two different ways: translating the non-binary problem into an equivalent binary one, or extending binary search algorithms to solve directly the original problem. The latter option raises some issues when we want to extend definitions written for the binary case. This paper focuses on the well-known forward checking algorithm, and shows that it can be generalized to several non-binary versions, all fitting its binary definition. The classical non-binary version, proposed by Van Hentenryck, is only one of these generalizations.

Current Approaches for Solving Over-Constrained Problems

We summarize existing approaches to model and solve overconstrained problems. These problems are usually formulated as combinatorial optimization problems, and different specific and generic formalisms are discussed, including the special case of multi-objective optimization. Regarding solving methods, both systematic and local search approaches are considered. Finally we review a number of case studies on overconstrained problems taken from the specialized literature.

Exploiting the use of DAC in MAX-CSP

Following the work of Wallace, who introduced the use of directed arc-consistency in MAX-CSP algorithms using DAC counts, we present a number of improvements of DAC usage for the P-EFC3 algorithm. These improvements include: (i) a better detection of dead-ends, (ii) a more effective form for value pruning, and (iii) a different heuristic criterion for value ordering. Considering the new DAC usage, we have analyzed some static variable ordering heuristics previously suggested, and we propose new ones which have been shown effective. The benefits of our proposal has been assessed empirically solving random CSP instances, showing a clear performance gain with respect to previous approaches.