Abstract
This paper deals with the challenging problem of counting the number of solutions of a CSP, denoted #CSP. Recent progress have been made using search methods, such as BTD [15], which exploit the constraint graph structure in order to solve CSPs. We propose to adapt BTD for solving the #CSP problem. The resulting exact counting method has a worst-case time complexity exponential in a specific graph parameter, called tree-width. For problems with sparse constraint graphs but large tree-width, we propose an iterative method which approximates the number of solutions by solving a partition of the set of constraints into a collection of partial chordal subgraphs. Its time complexity is exponential in the maximum clique size - the clique number - of the original problem, which can be much smaller than its tree-width. Experiments on CSP and SAT benchmarks shows the practical efficiency of our proposed approaches.
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References
Aceto, L., Hansen, J.A., Ingólfsdóttir, A., Johnsen, J., Knudsen, J.: The complexity of checking consistency of pedigree information and related problems. Journal of Computer Science Technology 19(1), 42–59 (2004)
Arnborg, S., Corneil, D., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM Journal of Discrete Mathematics 8, 277–284 (1987)
Bayardo, R., Pehoushek, J.: Counting models using connected components. In: AAAI 2000, pp. 157–162 (2000)
Choi, A., Darwiche, A.: An edge deletion semantics for belief propagation and its practical impact on approximation quality. In: Proc. of AAAI, pp. 1107–1114 (2006)
Darwiche, A.: On the tractable counting of theory models and its applications to truth maintenance and belief revision. Journal of Applied Non-classical Logic 11, 11–34 (2001)
Darwiche, A.: New advances in compiling cnf to decomposable negation normal form. In: Proc. of ECAI, pp. 328–332 (2004)
Dearing, P.M., Shier, D.R., Warner, D.D.: Maximal chordal subgraphs. Discrete Applied Mathematics 20(3), 181–190 (1988)
Dechter, R., Mateescu, R.: The impact of and/or search spaces on constraint satisfaction and counting. In: Proc. of CP, Toronto, CA, pp. 731–736 (2004)
Dechter, R., Mateescu, R.: And/or search spaces for graphical models. Artif. Intell. 171(2-3), 73–106 (2007)
Gogate, V., Dechter, R.: Approximate counting by sampling the backtrack-free search space. In: Proc. of AAAI 2007, Vancouver, CA, pp. 198–203 (2007)
Gogate, V., Dechter, R.: Approximate solution sampling ( and counting) on and/or search spaces. In: Proc. of CP 2008, Sydney, AU, pp. 534–538 (2008)
Gomes, C.P., Hoffmann, J., Sabharwal, A., Selman, B.: From sampling to model counting. In: Proc. of IJCAI, pp. 2293–2299 (2007)
Gomes, C.P., Sabharwal, A., Selman, B.: Model counting: A new strategy for obtaining good bounds. In: Proc. of AAAI-06, Boston, MA (2006)
Gomes, C.P., van Hoeve, W.-J., Sabharwal, A., Selman, B.: Counting CSP solutions using generalized XOR constraints. In: Proc. of AAAI 2007, Vancouver, BC, pp. 204–209 (2007)
Jégou, P., Terrioux, C.: Hybrid backtracking bounded by tree-decomposition of constraint networks. Artificial Intelligence 146, 43–75 (2003)
Kask, K., Dechter, R., Gogate, V.: New look-ahead schemes for constraint satisfaction. In: Proc. of AI&M (2004)
Kroc, L., Sabharwal, A., Selman, B.: Leveraging belief propagation, backtrack search, and statistics for model counting. In: Perron, L., Trick, M.A. (eds.) CPAIOR 2008. LNCS, vol. 5015, pp. 127–141. Springer, Heidelberg (2008)
Satish Kumar, T.K.: A model counting characterization of diagnoses. In: Proc. of the 13th International Workshop on Principles of Diagnosis (2002)
Lecoutre, C., Sais, L., Tabary, S., Vidal, V.: Last conflict based reasoning. In: Proc. of ECAI 2006, Trento, Italy, pp. 133–137 (2006)
Nishimura, N., Ragde, P., Szeider, S.: Solving #sat using vertex covers. Acta Inf. 44(7), 509–523 (2007)
Pesant, G.: Counting solutions of CSPs: A structural approach. In: Proc. of IJCAI, pp. 260–265 (2005)
Robertson, N., Seymour, P.D.: Graph minors II: Algorithmic aspects of tree-width. Algorithms 7, 309–322 (1986)
Roth, D.: On the hardness of approximate reasonning. Artificial Intelligence 82(1-2), 273–302 (1996)
Samer, M., Szeider, S.: A fixed-parameter algorithm for #sat with parameter incidence treewidth (2006)
Sanchez, M., de Givry, S., Schiex, T.: Mendelian error detection in complex pedigrees using weighted constraint satisfaction techniques. Constraints 13(1), 130–154 (2008)
Sang, T., Bacchus, F., Beame, P., Kautz, H., Pitassi, T.: Combining component caching and clause learning for effective model counting. In: SAT 2004, Vancouver, Canada (2004)
Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Sciences 8, 189–201 (1979)
Wei, W., Selman, B.: A new approach to model counting. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 324–339. Springer, Heidelberg (2005)
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Favier, A., de Givry, S., Jégou, P. (2009). Exploiting Problem Structure for Solution Counting. In: Gent, I.P. (eds) Principles and Practice of Constraint Programming - CP 2009. CP 2009. Lecture Notes in Computer Science, vol 5732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04244-7_27
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DOI: https://doi.org/10.1007/978-3-642-04244-7_27
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