Computational complexity of counting problems on 3-regular planar graphs

Xia, Mingji; Zhang, Peng; Zhao, Wenbo

doi:10.1016/j.tcs.2007.05.023

Cited by 56 publications

(36 citation statements)

References 13 publications

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“…. , s n t 0 ], for some s and t. The first problem is simply counting the number of vertex covers for 3-regular graphs; while the second is to count the number of (not necessarily perfect) matchings for 3-regular graphs [21]. We remark that both of them remain #P-Complete even for planar graphs.…”

Section: Interpolation Methodsmentioning

confidence: 99%

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Holographic reduction, interpolation and hardness

Cai

Xia

2012

comput. complex.

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We prove a dichotomy theorem for a class of counting problems expressible by Boolean signatures. The proof methods are holographic reductions and interpolations. We show that interpolatability provides a universal strategy to prove #P-hardness for this class of problems. For these problems whenever holographic reductions followed by interpolations fail to prove #P-hardness, we can show that the problems are actually solvable in polynomial time.

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Section: Interpolation Methodsmentioning

confidence: 99%

“…It is easy to verify that it is irreducible over Q[x], and by Lemma 5.1, we know that this family of gadgets can be used for interpolation. [21].…”

Section: The Hard Cases (Hard Even For Planar Graphs)mentioning

confidence: 99%

Holographic reduction, interpolation and hardness

Cai

Xia

2012

comput. complex.

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“…The unique flavour of modular counting is exhibited by Valiant's famous restricted version of 3-SAT [19] for which counting solutions is #P-complete [20], counting solutions modulo 7 is in polynomial-time but counting solutions modulo 2 is ⊕P-complete [19]. The seemingly mysterious number 7 was subsequently explained by Cai and Lu [4], who showed that the k-SAT version of Valiant's problem is tractable modulo any prime factor of 2 k − 1.…”

Section: Counting Modulomentioning

confidence: 99%

Counting Homomorphisms to Square-Free Graphs, Modulo 2

Göbel

Goldberg

Richerby

2016

ACM Trans. Comput. Theory

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We study the problem ⊕HomsToH of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph H. A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (non-modular) counting, so subtle dichotomy theorems can arise. We show the following dichotomy: for any H that contains no 4-cycles, ⊕HomsToH is either in polynomial time or is ⊕P-complete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs including graphs of unbounded tree-width. In particular, we focus on square-free graphs, which are graphs without 4-cycles. These graphs arise frequently in combinatorics, for example in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be tree-like so that tree-like decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.

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“…Cai and Lu [13] further showed that any matchgrid that is realizable in some basis is also realizable in a basis of size one. Holographic reductions have also been used to prove some new ⊕P-completeness [60] and #P-completeness [63] results.…”

Section: Note Added In July 2007mentioning

confidence: 99%

Holographic Algorithms

Valiant

2008

SIAM J. Comput.

192

155

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Abstract. Complexity theory is built fundamentally on the notion of efficient reduction among computational problems. Classical reductions involve gadgets that map solution fragments of one problem to solution fragments of another in one-to-one, or possibly one-to-many, fashion. In this paper we propose a new kind of reduction that allows for gadgets with many-to-many correspondences, in which the individual correspondences among the solution fragments can no longer be identified. Their objective may be viewed as that of generating interference patterns among these solution fragments so as to conserve their sum. We show that such holographic reductions provide a method of translating a combinatorial problem to finite systems of polynomial equations with integer coefficients such that the number of solutions of the combinatorial problem can be counted in polynomial time if one of the systems has a solution over the complex numbers. We derive polynomial time algorithms in this way for a number of problems for which only exponential time algorithms were known before. General questions about complexity classes can also be formulated. If the method is applied to a #P-complete problem, then polynomial systems can be obtained, the solvability of which would imply P #P = NC2.Key words. computational complexity, enumeration AMS subject classifications. 05A15, 68Q10, 68Q15, 68Q17, 68R10, 68W01DOI. 10.1137/0706825751. Introduction. Efficient reduction is perhaps the most fundamental notion on which the theory of computational complexity is built. The purpose of this paper is to introduce a new notion of efficient reduction, called a holographic reduction. In a classical reduction an instance of one problem is mapped to an instance of another by replacing its parts by certain gadgets. Solution fragments of the first problem will correspond in the gadgets to solution fragments of the second problem. For example, when mapping a Boolean satisfiability problem to a graph theory problem, each way of satisfying a part of the formula will correspond to a way of realizing a solution to the graph theory problem in the gadget. In classical reductions the correspondence between the solution fragments of the two problems is essentially oneto-one, or possibly many-to-one or one-to-many. In a holographic reduction the sum of the solution fragments of one problem maps to the sum of the solution fragments of the other problem for any one gadget and does so in such a way that the sum of all the overall solutions of the one will map to the sum of all the overall solutions of the other. The gadgets therefore map solution fragments many-to-many. The main innovation this allows is that it permits reductions in which correspondences between the solution fragments of the two problems need no longer be identifiable at all. Their effect can be viewed as that of producing interference patterns among the solution fragments, and they are called holographic gadgets for that reason.

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Computational complexity of counting problems on 3-regular planar graphs

Cited by 56 publications

References 13 publications

Holographic reduction, interpolation and hardness

Holographic reduction, interpolation and hardness

Counting Homomorphisms to Square-Free Graphs, Modulo 2

Holographic Algorithms

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