COCOON 2011: Computing and Combinatorics pp 122-133 | Cite as
Approximation Complexity of Complex-Weighted Degree-Two Counting Constraint Satisfaction Problems
Abstract
Constraint satisfaction problems have been studied in numerous fields with practical and theoretical interests. Recently, major breakthroughs have been made in the study of counting constraint satisfaction problems (or simply #CSPs). In particular, a computational complexity classification of bounded-degree #CSPs has been discovered for all degrees except for two, where the degree of an instance is the maximal number of times that each input variable appears in any given set of constraints. This paper challenges an open problem of classifying all degree-2 #CSPs on an approximate counting model and presents its partial solution by developing two novel proof techniques—T2-constructibility and parametrized symmetrization—which are specifically designed to handle arbitrary constraints under approximation-preserving reductions. Our proof exploits a close relationship between complex-weighted degree-2 #CSPs and Holant problems, which are a natural generalization of complex-weighted #CSPs.
Keywords
counting CSP bounded degree Holant problem signature AP-reducibility constructibility symmetrizationPreview
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References
- 1.Cai, J., Lu, P.: Holographic algorithms: from arts to science. J. Comput. Syst. Sci. 77, 41–61 (2011)MathSciNetCrossRefMATHGoogle Scholar
- 2.Cai, J., Lu, P., Xia, M.: Holographic algorithms by Fibonacci gates and holographic reductions for hardness. In: Proceedings of FOCS 2008, pp. 644–653 (2008)Google Scholar
- 3.Cai, J., Lu, P., Xia, M.: Holant problems and counting CSP. In: Proceedings of STOC 2009, pp. 715–724 (2009)Google Scholar
- 4.Cai, J., Lu, P., Xia, M.: Dichotomy for Holant* problems of Boolean domain, http://pages.cs.wisc.edu/~jyc/
- 5.Creignou, N., Hermann, M.: Complexity of generalized satisfiability counting problems. Inf. Comput. 125, 1–12 (1996)MathSciNetCrossRefMATHGoogle Scholar
- 6.Dalmau, V., Ford, D.K.: Generalized satisfiability with limited occurrences per variable: A study through delta-matroid parity. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 358–367. Springer, Heidelberg (2003)CrossRefGoogle Scholar
- 7.Dyer, M., Goldberg, L.A., Greenhill, C., Jerrum, M.: The relative complexity of approximating counting problems. Algorithmica 38, 471–500 (2003)CrossRefMATHGoogle Scholar
- 8.Dyer, M., Goldberg, L.A., Jalsenius, M., Richerby, D.: The complexity of approximating bounded-degree Boolean #CSP. In: Proceedings of STACS 2010, pp. 323–334 (2010)Google Scholar
- 9.Dyer, M., Goldberg, L.A., Jerrum, M.: The complexity of weighted Boolean #CSP. SIAM J. Comput. 38, 1970–1986 (2009)MathSciNetCrossRefMATHGoogle Scholar
- 10.Dyer, M., Goldberg, L.A., Jerrum, M.: An approximation trichotomy for Boolean #CSP. J. Comput. Syst. Sci. 76, 267–277 (2010)MathSciNetCrossRefMATHGoogle Scholar
- 11.Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of FOCS 1978, pp. 216–226 (1978)Google Scholar
- 12.Valiant, L.G.: Holographic algorithms. SIAM J. Comput. 37, 1565–1594 (2008)MathSciNetCrossRefMATHGoogle Scholar
- 13.Yamakami, T.: Approximate Counting for Complex-Weighted Boolean Constraint Satisfaction Problems. In: Jansen, K., Solis-Oba, R. (eds.) WAOA 2010. LNCS, vol. 6534, pp. 261–272. Springer, Heidelberg (2011); An improved version is available at arXive:1007.0391 CrossRefGoogle Scholar
- 14.Yamakami, T.: A Trichotomy Theorem for the Approximate Counting of Complex-Weighted Bounded-Degree Boolean CSPs. In: Wu, W., Daescu, O. (eds.) COCOA 2010, Part I. LNCS, vol. 6508, pp. 285–299. Springer, Heidelberg (2010); An improved version is available at arXiv:1008.2688 CrossRefGoogle Scholar