Gadgets and anti-gadgets leading to a complexity dichotomy

Cai, Jin-Yi; Kowalczyk, Michael; Williams, Tyson

doi:10.1145/2090236.2090272

Cited by 17 publications

(21 citation statements)

References 38 publications

(62 reference statements)

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“…This theme is generally proved for symmetric signatures [15,9,13]. For not-necessarily-symmetric signatures, these are only proved in special cases [4]. This paper provides a firm foundation for this theory and for future explorations.…”

Section: Resultsmentioning

confidence: 87%

Untitled

Cai¹,

Gorenstein²

2014

Theory of Comput.

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Section: Resultsmentioning

confidence: 87%

Untitled

Cai¹,

Gorenstein²

2014

Theory of Comput.

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“…Thus there exists h ∈ H 2 \ M . If there exists g ∈ G 2 such that g ∈ M † \ (P ∪ A ), then Pl-#CSP 2 (f, g) is #P-hard by Theorem F. 9. Otherwise, G 2 ⊆ M † ∩ (P ∪ A ).…”

Section: G No-mixing Of Even Arity Signature Setmentioning

confidence: 99%

“…Definition 2. 9. Let S n be the symmetric group of degree n. Then for positive integers t and n with t ≤ n and unary signatures v, v 1 , .…”

Section: Tractable Signature Setsmentioning

confidence: 99%

“…Also note that P and A are invariant under T , and since g ′ ∈ M † \(P ∪ A ), we haveĝ ′ ∈ M \(P ∪ A ). In the following, we will construct [ Theorem F. 9. Let f and g be two symmetric signatures of even arity.…”

Section: Suppose Gmentioning

confidence: 99%

“…These date back to the classification of the complexity of the Tutte polynomial [41,40]. It has also been an unfailing theme in the classification of spin systems and #CSP [25,12,9,20]. However, these frameworks do not capture all locally specified counting problems.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Holant Dichotomy: Is the FKT Algorithm Universal?

Cai

Guo

et al. 2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specifically to answer the question: Is the FKT algorithm under a holographic transformation [38] a universal strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable in general? This dichotomy is a culmination of previous ones, including those for Spin Systems [25], Holant [21,6], and #CSP [20].In the study of counting complexity, such as #CSP, there are problems which are #P-hard over general graphs but polynomial-time solvable over planar graphs. A recurring theme has been that a holographic reduction to FKT precisely captures these problems. Surprisingly, for planar Holant, we discover new planar tractable problems that are not expressible by a holographic reduction to FKT. In particular, a straightforward formulation of a dichotomy for planar Holant problems along the above recurring theme is false.In previous work, an important tool was a dichotomy for #CSP d , which denotes #CSP where every variable appears a multiple of d times. However the very first step in the #CSP d dichotomy proof fundamentally violates planarity. In fact, due to our newly discovered tractable problems, the putative form of a planar #CSP d dichotomy is false when d ≥ 5. Nevertheless, we prove a dichotomy for planar #CSP 2 . In this case, the putative form of the dichotomy is true. We manage to prove the planar Holant dichotomy without relying on a planar #CSP d dichotomy for d ≥ 3, while the dichotomy for planar #CSP 2 plays an essential role.As a special case of our new planar tractable problems, counting perfect matchings (#PM) over k-uniform hypergraphs is polynomial-time computable when the incidence graph is planar and k ≥ 5. The same problem is #P-hard when k = 3 or k = 4, which is also a consequence of our dichotomy. When k = 2, it becomes #PM over planar graphs and is tractable again. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is polynomial-time computable if the greatest common divisor (gcd) of all hyperedge sizes is at least 5. It is worth noting that it is the gcd, and not a bound on hyperedge sizes, that is the criterion for tractability.

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The Complexity of Planar Boolean #CSP with Complex Weights

Guo

Williams

2013

Automata, Languages, and Programming

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We prove a complexity dichotomy theorem for symmetric complex-weighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #P-hard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also obtain a dichotomy theorem for a symmetric arity 4 signature with complex weights in the planar Holant framework, which we use in the proof of our #CSP dichotomy. In particular, we reduce the problem of evaluating the Tutte polynomial of a planar graph at the point (3, 3) to counting Eulerian orientations over planar 4-regular graphs to show the latter is #P-hard. This strengthens a theorem by Huang and Lu to the planar setting. Our proof techniques combine new ideas with refinements and extensions of existing techniques. These include planar pairings, the unary recursive construction, the anti-gadget technique, and pinning in the Hadamard basis.

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Gadgets and anti-gadgets leading to a complexity dichotomy

Cited by 17 publications

References 38 publications

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A Holant Dichotomy: Is the FKT Algorithm Universal?

The Complexity of Planar Boolean #CSP with Complex Weights

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