Counting Hypergraph Matchings up to Uniqueness Threshold

Song, Renjie; Yin, Yitong; Zhao, Jinman

doi:10.4230/lipics.approx-random.2016.46

Approximating partition functions of bounded-degree Boolean counting Constraint Satisfaction Problems

Galanis

¹

,

Goldberg

²

,

Yang

³

2021

Journal of Computer and System Sciences

2

0

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We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language Γ and a degree bound ∆, we study the complexity of #CSP ∆ (Γ), which is the problem of counting satisfying assignments to CSP instances with constraints from Γ and whose variables can appear at most ∆ times. Our main result shows that: (i) if every function in Γ is affine, then #CSP ∆ (Γ) is in FP for all ∆, (ii) otherwise, if every function in Γ is in a class called IM 2 , then for all sufficiently large ∆, #CSP ∆ (Γ) is equivalent under approximation-preserving (AP) reductions to the counting problem #BIS (the problem of counting independent sets in bipartite graphs) (iii) otherwise, for all sufficiently large ∆, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSP ∆ (Γ), even within an exponential factor. Our result extends previous results, which apply only in the so-called "conservative" case.

show abstract

Approximating partition functions of bounded-degree Boolean counting Constraint Satisfaction Problems

Galanis

¹

,

Goldberg

²

,

Yang

³

2016

Preprint

0

View full text Add to dashboard Cite

We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language Γ and a degree bound ∆, we study the complexity of #CSP ∆ (Γ), which is the problem of counting satisfying assignments to CSP instances with constraints from Γ and whose variables can appear at most ∆ times. Our main result shows that: (i) if every function in Γ is affine, then #CSP ∆ (Γ) is in FP for all ∆, (ii) otherwise, if every function in Γ is in a class called IM 2 , then for all sufficiently large ∆, #CSP ∆ (Γ) is equivalent under approximation-preserving (AP) reductions to the counting problem #BIS (the problem of counting independent sets in bipartite graphs) (iii) otherwise, for all sufficiently large ∆, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSP ∆ (Γ), even within an exponential factor. Our result extends previous results, which apply only in the so-called "conservative" case.

show abstract