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Combining restarts, nogoods and bag-connected decompositions for solving CSPs

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Abstract

From a theoretical viewpoint, the (tree-)decomposition methods offer a good approach for solving Constraint Satisfaction Problems (CSPs) when their (tree)-width is small. In this case, they have often shown their practical interest. So, the literature (coming from Mathematics, OR or AI) has concentrated its efforts on the minimization of a single parameter, namely the tree-width. Nevertheless, experimental studies have shown that this parameter is not always the most relevant to consider when solving CSPs. So, in this paper, we highlight two fundamental problems related to the use of tree-decomposition and for which we offer two particularly appropriate solutions. First, we experimentally show that the decomposition algorithms of the state of the art produce clusters (a tree-decomposition is a rooted tree of clusters) having several connected components. We highlight the fact that such clusters create a real disadvantage which affects significantly the efficiency of solving methods. To avoid this problem, we consider here a new graph decomposition called Bag-Connected Tree-Decomposition, which considers only tree-decompositions such that each cluster is connected. We analyze such decompositions from an algorithmic point of view, especially in order to propose a first polynomial time algorithm to compute them. Moreover, even if we consider a very well suited decomposition, it is well known that sometimes, a bad choice for the root cluster may significantly degrade the performance of the solving. We highlight an explanation of this degradation and we propose a solution based on restart techniques. Then, we present a new version of the BTD algorithm (for Backtracking with Tree-Decomposition Jégou and Terrioux, Artificial Intelligence, 146 43–75 28) integrating restart techniques. From a theoretical viewpoint, we prove that reduced nld-nogoods can be safely recorded during the search and that their size is smaller than ones recorded by MAC+RST+NG (Lecoutre et al., JSAT, 1(3–4) 147–167 34). We also show how structural (no)goods may be exploited when the search restarts from a new root cluster. Finally, from a practical viewpoint, we show experimentally the benefits of using independently bag-connected tree-decompositions and restart techniques for solving CSPs by decomposition methods. Above all, we experimentally highlight the advantages brought by exploiting jointly these improvements in order to respond to two major problems generally encountered when solving CSPs by decomposition methods.

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Notes

  1. We use the term “bag” rather than “cluster” because it is more compatible with the terminology of Graph Theory.

  2. We assume that E i ∩E p(i) = ∅ if E i is the root cluster.

  3. See http://www.cril.univ-artois.fr/CPAI08 for more details.

  4. For any Y⊆X, the subgraph G[Y] of G=(X,C) induced by Y is the graph (Y,C Y ) where C Y ={{x,y}∈C|x,y∈Y}.

  5. Note that we use the term of Bag-Connected Tree-Width rather than one of Connected Tree-Width exploited in [36] because the term of Connected Tree-Width has been introduced before in [15] but corresponds to a quite different concept.

  6. 6 A cycle is said geodesic if for any pair of vertices x and y belonging to the cycle, the distance between x and y in the graph is equal to the length of the shortest path between x and y in the cycle.

  7. 7 Note that this hardware configuration is very different from one used in [29, 30].

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Acknowledgments

This work was supported by the French National Research Agency under grant TUPLES (ANR-2010-BLAN-0210). The authors would like to thank Ioan Todinca for their fruitful discussion about this work.

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Authors and Affiliations

  1. Aix Marseille Université, CNRS, ENSAM, Université de Toulon, LSIS UMR 7296, 13397, Marseille, France

    Philippe Jégou & Cyril Terrioux

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  1. Philippe Jégou
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  2. Cyril Terrioux
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Correspondence to Cyril Terrioux.

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This paper is an extension of the works published in [29, 30].

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Jégou, P., Terrioux, C. Combining restarts, nogoods and bag-connected decompositions for solving CSPs. Constraints 22, 191–229 (2017). https://doi.org/10.1007/s10601-016-9248-8

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