The complexity of complex weighted Boolean #CSP

Cai, Jin-Yi; Lu, Pinyan; Xia, Mingji

doi:10.1016/j.jcss.2013.07.003

Cited by 58 publications

(100 citation statements)

References 23 publications

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“…Therefore both the edge weight and the vertex weight of Factor-K 2 -NormIQPIsing(2θ) are powers of i. The algorithm from [7] (affine-type) can be used to solve Factor-K 2 -NormIQPIsing(2θ). See also case 1 of Theorem 6.…”

Section: Iqp and The Ising Partition Functionmentioning

confidence: 99%

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The Complexity of Approximating complex-valued Ising and Tutte partition functions

Goldberg

Guo

2017

comput. complex.

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We study the complexity of approximately evaluating the Ising and Tutte partition functions with complex parameters. Our results are partly motivated by the study of the quantum complexity classes BQP and IQP. Recent results show how to encode quantum computations as evaluations of classical partition functions. These results rely on interesting and deep results about quantum computation in order to obtain hardness results about the difficulty of (classically) evaluating the partition functions for certain fixed parameters.The motivation for this paper is to study more comprehensively the complexity of (classically) approximating the Ising and Tutte partition functions with complex parameters. Partition functions are combinatorial in nature and quantifying their approximation complexity does not require a detailed understanding of quantum computation. Using combinatorial arguments, we give the first full classification of the complexity of multiplicatively approximating the norm and additively approximating the argument of the Ising partition function for complex edge interactions (as well as of approximating the partition function according to a natural complex metric). We also study the norm approximation problem in the presence of external fields, for which we give a complete dichotomy when the parameters are roots of unity. Previous results were known just for a few such points, and we strengthen these results from BQP-hardness to #P-hardness. Moreover, we show that computing the sign of the Tutte polynomial is #P-hard at certain points related to the simulation of BQP. Using our classifications, we then revisit the connections to quantum computation, drawing conclusions that are a little different from (and incomparable to) ones in the quantum literature, but along similar lines.

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Section: Iqp and The Ising Partition Functionmentioning

confidence: 99%

“…Consider next the case where y = ±i. If z ∈ {1, −1, i, −i}, the algorithm is from [7]. Otherwise, the hardness is from Lemma 46.…”

Section: −M Ymentioning

confidence: 99%

The Complexity of Approximating complex-valued Ising and Tutte partition functions

Goldberg

Guo

2017

comput. complex.

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“…By the dichotomy theorem for #CSP with each variable appearing at most three times [12], we know that the problem is #P-hard…”

Section: Proof Outlinementioning

confidence: 99%

“…For #CSP over the Boolean domain, two ex-plicit tractable families, namely P (product type) and A (affine type), are identified; any function set not contained in these two families is proved to be #P-hard. The result was first proved for unweighted 0-1 valued constraint functions [14], later for non-negatively weighted functions [15], and finally for complex valued functions [12]. From non-negative values to complex values, the tractable family A expands highly non-trivially; the tractability incorporates cancelations and the proof depends on a nice algebraic structure.…”

Section: Introductionmentioning

confidence: 99%

Dichotomy for Real Holant^c Problems

Cai

Xia

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Holant problems capture a class of Sum-of-Product computations such as counting matchings. It is inspired by holographic algorithms and is equivalent to tensor networks, with counting CSP being a special case. A complexity classification for Holant problems is more difficult to prove, not only because it logically implies a classification for counting CSP, but also due to the deeper reason that there exist more intricate polynomial time tractable problems in the broader framework. We discover a new family of constraint functions L which define polynomial time computable counting problems. These do not appear in counting CSP, and no newly discovered tractable constraints can be symmetric. It has a delicate support structure related to error-correcting codes. Local holographic transformations is fundamental in its tractability. We prove a complexity dichotomy theorem for all Holant problems defined by any real valued constraint function set on Boolean variables and contains two 0-1 pinning functions. Previously, dichotomy for the same framework was only known for symmetric constraint functions. The set L supplies the last piece of tractability. We also prove a dichotomy for a variant of counting CSP as a technical component toward this Holant dichotomy.

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“…Dyer, Goldberg and Jerrum [19] extended Creignou and Hermann's result to weighted Boolean #CSP. Cai, Lu and Xia [6,7] extended further to the case of complex weights and show that the dichotomy holds for the restriction of the problem in which instances have degree 3. Their result implies that the degree-3 problem #CSP 3 (Γ ) (#CSP(Γ ) restricted to instances of degree 3) has a polynomial-time algorithm if every relation in Γ is affine and is #P-complete, otherwise.…”

Section: Counting Cspmentioning

confidence: 99%

The complexity of approximating bounded-degree Boolean #CSP

Dyer

Goldberg

Jalsenius

et al. 2012

Information and Computation

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The degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP = RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs.

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The complexity of complex weighted Boolean #CSP

Cited by 58 publications

References 23 publications

The Complexity of Approximating complex-valued Ising and Tutte partition functions

The Complexity of Approximating complex-valued Ising and Tutte partition functions

Dichotomy for Real Holant^c Problems

The complexity of approximating bounded-degree Boolean #CSP

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The complexity of complex weighted Boolean #CSP

Cited by 58 publications

References 23 publications

The Complexity of Approximating complex-valued Ising and Tutte partition functions

The Complexity of Approximating complex-valued Ising and Tutte partition functions

Dichotomy for Real Holantc Problems

The complexity of approximating bounded-degree Boolean #CSP

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Dichotomy for Real Holant^c Problems