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Holant problems and counting CSP

Published: 31 May 2009 Publication History

Abstract

We propose and explore a novel alternative framework to study the complexity of counting problems, called Holant Problems. Compared to counting Constrained Satisfaction Problems (CSP), it is a refinement with a more explicit role for the function constraints. Both graph homomorphism and CSP can be viewed as special cases of Holant Problems. We prove complexity dichotomy theorems in this framework. Because the framework is more stringent, previous dichotomy theorems for CSP problems no longer apply. Indeed, we discover surprising tractable subclasses of counting problems, which could not have been easily specified in the CSP framework. The main technical tool we use and develop is holographic reductions. Another technical tool used in combination with holographic reductions is polynomial interpolations. The study of Holant Problems led us to discover and prove a complexity dichotomy theorem for the most general form of Boolean CSP where every constraint function takes values in the complex number field {C}.

References

[1]
Andrei A. Bulatov. A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM, 53(1):66--120, 2006.
[2]
Andrei A. Bulatov. The complexity of the counting constraint satisfaction problem. In ICALP (1), volume 5125 of Lecture Notes in Computer Science, pages 646--661. Springer, 2008.
[3]
Andrei A. Bulatov and V'ıctor Dalmau. Towards a dichotomy theorem for the counting constraint satisfaction problem. Inf. Comput., 205(5):651--678, 2007.
[4]
Andrei A. Bulatov and Martin Grohe. The complexity of partition functions. In ICALP, volume 3142 of Lecture Notes in Computer Science, pages 294--306. Springer, 2004.
[5]
Andrei A. Bulatov and Martin Grohe. The complexity of partition functions. Theor. Comput. Sci., 348(2-3):148--186, 2005.
[6]
Jin-Yi Cai, Xi Chen, and Pinyan Lu. Graph homomorphisms with complex values: A dichotomy theorem. manuscript, 2009.
[7]
Jin--Yi Cai and Pinyan Lu. Holographic algorithms: from art to science. In STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 401--410, 2007.
[8]
Jin-Yi Cai and Pinyan Lu. On symmetric signatures in holographic algorithms. In STACS, volume 4393 of Lecture Notes in Computer Science, pages 429--440. Springer, 2007.
[9]
Jin-Yi Cai, Pinyan Lu, and Mingji Xia. Holographic algorithms by fibonacci gates and holographic reductions for hardness. In FOCS '08: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, pages 644--653, 2008.
[10]
N. Creignou, S. Khanna, and M. Sudan. Complexity classifications of boolean constraint satisfaction problems. SIAM Monographs on Discrete Mathematics and Applications, 2001.
[11]
P. Dagum and M. Luby. Approximating the permanent of graphs with large factors. Theor. Comput. Sci., 102:283--305, 1992.
[12]
C. T. J. Dodson and T. Poston. Tensor Geometry. Graduate Texts in Mathematics 130. Springer-Verlag, New York, 1991.
[13]
Martin E. Dyer, Leslie Ann Goldberg, and Mark Jerrum. The complexity of weighted boolean CSP. CoRR, abs/0704.3683, 2007.
[14]
Martin E. Dyer, Leslie Ann Goldberg, and Mike Paterson. On counting homomorphisms to directed acyclic graphs. In ICALP (1), volume 4051 of Lecture Notes in Computer Science, pages 38--49. Springer, 2006.
[15]
Martin E. Dyer, Leslie Ann Goldberg, and Mike Paterson. On counting homomorphisms to directed acyclic graphs. J. ACM, 54(6), 2007.
[16]
Martin E. Dyer and Catherine S. Greenhill. The complexity of counting graph homomorphisms (extended abstract). In SODA, pages 246--255, 2000.
[17]
Martin E. Dyer and Catherine S. Greenhill. Corrigendum: The complexity of counting graph homomorphisms. Random Struct. Algorithms, 25(3):346--352, 2004.
[18]
Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic snp and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput., 28(1):57--104, 1998.
[19]
M. Freedman, L. Lovász, and A. Schrijver. Reflection positivity, rank connectivity, and homomorphism of graphs. J. AMS, 20:37--51, 2007.
[20]
Leslie Ann Goldberg, Martin Grohe, Mark Jerrum, and Marc Thurley. A complexity dichotomy for partition functions with mixed signs. CoRR, abs/0804.1932, 2008.
[21]
P. Hell and J. Nesetril. On the complexity of h-coloring. Journal of Combinatorial Theory, Series B, 48(1):92--110, 1990.
[22]
P. W. Kasteleyn. The statistics of dimers on a lattice. Physica, 27:1209--1225, 1961.
[23]
P. W. Kasteleyn. Graph theory and crystal physics. In Graph Theory and Theoretical Physics, pages 43--110. Academic Press, London, 1967.
[24]
H. N. V. Temperley and M. E. Fisher. Dimer problem in statistical mechanics -- an exact result. Philosophical Magazine, 6:1061--1063, 1961.
[25]
Salil P. Vadhan. The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput., 31(2):398--427, 2001.
[26]
Leslie G. Valiant. The complexity of enumeration and reliability problems. SIAM J. Comput., 8(3):410--421, 1979.
[27]
Leslie G. Valiant. Holographic algorithms (extended abstract). In FOCS '04: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pages 306--315,2004.
[28]
Leslie G. Valiant. Accidental algorthims. In FOCS '06: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pages 509--517, 2006.
[29]
Mingji Xia, Peng Zhang, and Wenbo Zhao. Computational complexity of counting problems on 3-regular planar graphs. Theor. Comput. Sci., 384(1):111--125, 2007.

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    cover image ACM Conferences
    STOC '09: Proceedings of the forty-first annual ACM symposium on Theory of computing
    May 2009
    750 pages
    ISBN:9781605585062
    DOI:10.1145/1536414
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    Published: 31 May 2009

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    Author Tags

    1. CSP
    2. holant problem
    3. holographic reduction
    4. polynomial interpolation

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    STOC '09: Symposium on Theory of Computing
    May 31 - June 2, 2009
    MD, Bethesda, USA

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