research-article

Zero-freeness and approximation of real Boolean Holant problems

Authors:

Hanpin WangAuthors Info & Claims

Volume 917, Issue C

Pages 12 - 30

https://doi.org/10.1016/j.tcs.2022.03.009

Published: 25 May 2022 Publication History

Highlights

•

Show the complexity of approximating real Boolean Holant problems.

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Present holographic transformations from the Holant problems to the Ising model.

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Prove zero-freeness of the Holant problems.

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Establish AP-reductions between the Holant problems and counting perfect matchings.

Abstract

Holant problems provide a novel framework to study the complexity of counting problems. It is a refinement to counting constraint satisfaction problems (#CSP) with a more explicit role for the constraint functions. Both graph homomorphisms and #CSP can be viewed as special cases of Holant problems. For approximation algorithms on Holant problems, the attention is focused on proving zero-freeness of the partition functions and establishing fully polynomial-time approximation schemes (FPTAS) using the Taylor expansion method. In this paper, we study the Holant problems defined by a real constraint function satisfying a generalized second-order recurrence. We present fully polynomial-time (deterministic or randomized) approximation schemes for the Holant problems except for a couple of cases. Our algorithms are established in two ways: 1) we construct holographic transformations from the Holant problems to the Ising model, and obtain the algorithms using the approaches with respect to the Ising model; 2) we prove the zero-freeness of the Holant problems, and present an algorithm based on the Taylor expansion method. In addition, for most of the other cases, there exist approximation-preserving reductions between the Holant problems and the problem of counting perfect matchings, which is a central open problem in approximate counting.

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Index Terms

Zero-freeness and approximation of real Boolean Holant problems
1. Mathematics of computing
  1. Discrete mathematics
  2. Mathematical analysis
    1. Functional analysis
      1. Approximation
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

Index terms have been assigned to the content through auto-classification.

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Published In

cover image Theoretical Computer Science

Theoretical Computer Science Volume 917, Issue C

May 2022

122 pages

ISSN:0304-3975

Issue’s Table of Contents

Elsevier B.V.

Publisher

Elsevier Science Publishers Ltd.

United Kingdom

Publication History

Published: 25 May 2022

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