Romuald Debruyne

Domain filtering consistencies

Enforcing local consistencies is one of the main features of constraint reasoning. Which level of local consistency should be used when searching for solutions in a constraint network is a basic question. Arc consistency and partial forms of arc consistency have been widely studied, and have been known for sometime through the forward checking or the MAC search algorithms. Until recently, stronger forms of local consistency remained limited to those that change the structure of the constraint graph, and thus, could not be used in practice, especially on large networks. This paper focuses on the local consistencies that are stronger than arc consistency, without changing the structure of the network, i.e., only removing inconsistent values from the domains. In the last five years, several such local consistencies have been proposed by us or by others. We make an overview of all of them, and highlight some relations between them. We compare them both theoretically and experimentally, considering their pruning efficiency and the time required to enforce them.

Maintaining Arc-Consistency within Dynamic Backtracking

Most of complete search algorithms over Constraint Satisfaction Problems (csp) are based on Standard Backtracking. Two main enhancements of this basic scheme have been studied: first, to integrate constraint propagation as mac which maintains arc consistency during search; second, intelligent backtrackers which avoid repeatedly falling in the same dead-ends by recording nogoods as Conflict-directed BackJumping (cbj) or Dynamic Backtracking (dbt). Integrations of constraint propagation within intelligent backtrackers have been done as mac-cbj which maintains arc consistency in cbj. However, Bessiere and REgin have shown that mac-cbj was very rarely better than mac. However, the inadequacy of mac-cbj is more related to the fact that cbj does not avoid thrashing than to the cost of the management of nogoods. This paper describes and evaluates mac-dbt which maintains arc-consistency in dbt. Experiments show that mac-dbt is able to solve very large problems and that it remains very stable as the size of the problems increases. Moreover, mac-dbt outperforms mac on the structured problems we have randomly generated.

From restricted path consistency to Max-Restricted Path Consistency

There is no need to show the importance of the filtering techniques to solve constraint satisfaction problems i.e. to find values for problem variables subject to constraints that specify which combinations of values are consistent. They can be used during a preprocessing step to remove once and for all some local inconsistencies, or during the search to efficiently prune the search tree. Recently, in [5], a comparison of the most practicable filtering techniques concludes that restricted path consistency (RPC) is a promising local consistency that requires little additional cpu time compared to arc consistency while removing most of the path inverse inconsistent values. However, the RPC algorithm used for this comparison (presented in [1] and called RPC1 in the following) has a non optimal worst case time complexity and bad average time and space complexities. Therefore, we propose RPC2, a new RPC algorithm with O(end2) worst case time complexity and requiring less space than RPC1 in practice. The second aim of this paper is to extend RPC to new local consistencies, k-RPC and Max-RPC, and to compare their pruning efficiency with the other practicable local consistencies. Furthermore, we propose and study a Max-RPC algorithm based on AC-6 that we used for this comparison.

Theoretical analysis of singleton arc consistency and its extensions

Singleton arc consistency (SAC) is a consistency property that is simple to specify and is stronger than arc consistency. Algorithms have already been proposed to enforce SAC, but they have a high time complexity. In this paper, we give a lower bound to the worst-case time complexity of enforcing SAC on binary constraints. We discuss two interesting features of SAC. The first feature leads us to propose an algorithm for SAC that has optimal time complexity when restricted to binary constraints. The second feature leads us to extend SAC to a stronger level of local consistency that we call Bidirectional SAC (BiSAC). We also show the relationship between SAC and the propagation of disjunctive constraints.

Efficient algorithms for singleton arc consistency

In this paper, we propose two original and efficient approaches for enforcing singleton arc consistency. In the first one, the data structures used to enforce arc consistency are shared between all subproblems where a domain is reduced to a singleton. This new algorithm is not optimal but it requires far less space and is often more efficient in practice than the optimal algorithm SAC-Opt. In the second approach, we perform several runs of a greedy search (where at each step, arc consistency is maintained), possibly detecting the singleton arc consistency of several values in one run. It is an original illustration of applying inference (i.e., establishing singleton arc consistency) by search. Using a greedy search allows benefiting from the incrementality of arc consistency, learning relevant information from conflicts and, potentially finding solution(s) during the inference process. We present extensive experiments that show the benefit of our two approaches.

Reformulation of Global Constraints Based on Constraints Checkers

This article deals with global constraints for which the set of solutions can be recognized by an extended finite automaton whose size is bounded by a polynomial in n, where n is the number of variables of the corresponding global constraint. By reducing the automaton to a conjunction of signature and transition constraints we show how to systematically obtain an automaton reformulation. Under some restrictions on the signature and transition constraints, this reformulation maintains arc-consistency. An implementation based on some constraints as well as on the metaprogramming facilities of SICStus Prolog is available. For a restricted class of automata we provide an automaton reformulation for the relaxed case, where the violation cost is the minimum number of variables to unassign in order to get back to a solution.