Combining restarts, nogoods and bag-connected decompositions for solving CSPs
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.
KeywordsConstraint satisfaction problems Tree-decomposition Bag-connected tree-decomposition Restart
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.
- 2.Berge, C. (1973). Graphs and Hypergraphs. Elsevier.Google Scholar
- 4.Bessière, C., & Régin, J.C. (2001). Refining the basic constraint propagation algorithm. In Proceedings of IJCAI (pp. 309–315).Google Scholar
- 5.Boussemart, F., Hemery, F., Lecoutre, C., & Sais, L. (2004). Boosting systematic search by weighting constraints. In Proceedings of ECAI (pp. 146–150).Google Scholar
- 7.Dechter, R. (2003). Constraint processing. Morgan Kaufmann Publishers.Google Scholar
- 9.Dechter, R., & Mateescu, R. (2004). The impact of AND/OR search spaces on constraint satisfaction and counting. In Proceedings of the 10th international conference on principles and practice of constraint programming (CP) (pp. 731–736).Google Scholar
- 12.Dermaku, A., Ganzow, T., Gottlob, G., McMahan, B.J., Musliu, N., & Samer, M. (2008). Heuristic methods for hypertree decomposition. In Proceedings of MICAI (pp. 1–11).Google Scholar
- 13.Diestel, R., & Müller, M. (2014). Connected tree-width. arXiv:1211.7353v2.
- 14.Favier, A, de Givry, S., & Jégou, P. (2009). Exploiting problem structure for solution counting. In Proceedings of CP (pp. 335–343).Google Scholar
- 15.Fraigniaud, P., & Nisse, N. (2006). Connected treewidth and connected graph searching. In Proceedings of LATIN (pp. 479–490).Google Scholar
- 22.Hamann, M., & Weißauer, D. (2015). Bounding connected tree-width. ArXiv:1503.01592.
- 23.Harvey, W.D. (1995). Nonsystematic backtracking search. Ph.D. thesis, Stanford University.Google Scholar
- 24.Jégou, P., Ndiaye, S.N., & Terrioux, C. (2005). Computing and exploiting tree-decompositions for solving constraint networks. In Proceedings of CP (pp. 777–781).Google Scholar
- 25.Jégou, P., Ndiaye, S.N., & Terrioux, C. (2006). An extension of complexity bounds and dynamic heuristics for tree-decompositions of CSP. In Proceedings of CP (pp. 741–745).Google Scholar
- 26.Jégou, P., Ndiaye, S.N., & Terrioux, C. (2007). Dynamic heuristics for backtrack search on tree-decomposition of CSPs. In Proceedings of IJCAI (pp. 112–117).Google Scholar
- 27.Jégou, P., Ndiaye, S.N., & Terrioux, C. (2007). Dynamic management of heuristics for solving structured CSPs. In Proceedings of CP (pp. 364–378).Google Scholar
- 29.Jėgou, P., & Terrioux, C. (2014). Combining restarts, nogoods and decompositions for solving csps. In Proceedings of ECAI (pp. 465–470).Google Scholar
- 30.Jėgou, P., & Terrioux, C. (2014). Tree-decompositions with connected clusters for solving constraint networks. In Proceedings of CP (pp. 407–423).Google Scholar
- 31.Karakashian, S., Woodward, R., & Choueiry, B.Y. (2013). Improving the performance of consistency algorithms by localizing and bolstering propagation in a tree decomposition. In Proceedings of AAAI (pp. 466–473).Google Scholar
- 32.Kjaerulff, U. (1990). Triangulation of graphs - algorithms giving small total state space. Tech. rep., Judex R.R. Aalborg, Denmark.Google Scholar
- 33.Lecoutre, C. (2009). Constraint networks - techniques and algorithms. ISTE/Wiley.Google Scholar
- 36.Müller, M. (2012). Connected tree-width. ArXiv:1211.7353.
- 37.Nadel, B. (1988). Tree search and arc consistency in constraint-satisfaction algorithms. In Search in artificial intelligence (pp. 287–342). Springer-Verlag.Google Scholar
- 38.Petke, J. (2012). On the bridge between constraint satisfaction and Boolean satisfiability. Ph.D. thesis, University of Oxford.Google Scholar
- 40.Rose, D.J. (1973). A graph theoretic study of the numerical solution of sparse positive denite systems of linear equations. In Read, R.C. (Ed.) Graph theory and computing (pp. 183–217). New York: Academic Press.Google Scholar
- 41.Rossi, F., van Beek, P., & Walsh, T. (2006). Handbook of constraint programming. Elsevier.Google Scholar
- 42.Sabin, D., & Freuder, E. (1994). Contradicting conventional wisdom in constraint satisfaction. In Proceedings of ECAI (pp. 125–129).Google Scholar
- 43.Sabin, D., & Freuder, E. (1997). Understanding and improving the MAC algorithm. In Proceedings of CP (pp. 167–181).Google Scholar
- 44.Sanchez, M., Bouveret, S, de Givry, S., Heras, F., Jégou, P., Larrosa, J., Ndiaye, S.N., Rollon, E., Schiex, T., Terrioux, C., Verfaillie, G., & Zytnicki, M. (2008). Max-CSP competition 2008: toulbar2 solver description. In Proceedings of the 3rd CSP solver competition, CP workshop (pp. 63–70).Google Scholar
- 46.Walsh, T. (1999). Search in a small world. In Proceedings of IJCAI (pp. 1172–1177).Google Scholar