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

We study the complexity of #CSP∆(&Gamma;), which is the problem of counting satisfying assignments to CSP instances with constraints from &Gamma; and whose variables can appear at most ∆ times. Our main result shows that: (i) if every function in &Gamma; is affine, then #CSP∆(&Gamma;) is in FP for all ∆, (ii) otherwise, if every function in &Gamma; is in a class called IM2, then for large ∆, #CSP∆(&Gamma;) is equivalent under approximation-preserving reductions to the problem of counting independent sets in bipartite graphs, (iii) otherwise, for large ∆, it is NP-hard to approximate #CSP∆(&Gamma;), even within an exponential factor.</p

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

We study the complexity of #CSP∆(&Gamma;), which is the problem of counting satisfying assignments to CSP instances with constraints from &Gamma; and whose variables can appear at most ∆ times. Our main result shows that: (i) if every function in &Gamma; is affine, then #CSP∆(&Gamma;) is in FP for all ∆, (ii) otherwise, if every function in &Gamma; is in a class called IM2, then for large ∆, #CSP∆(&Gamma;) is equivalent under approximation-preserving reductions to the problem of counting independent sets in bipartite graphs, (iii) otherwise, for large ∆, it is NP-hard to approximate #CSP∆(&Gamma;), even within an exponential factor.</p

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

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 Δ, we study the complexity of #CSPΔ(Γ), which is the problem of counting satisfying assignments to CSP instances with constraints from Γ and whose variables can appear at most Δ times. Our main result shows that: (i) if every function in Γ is affine, then #CSPΔ(Γ) is in FP for all Δ, (ii) otherwise, if every function in Γ is in a class called IM2, then for all sufficiently large Δ, #CSPΔ(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 Δ, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSPΔ(Γ), even within an exponential factor. Our result extends previous results, which apply only in the so-called “conservative” case. A full version of the paper containing detailed proofs is available at http://arxiv.org/abs/1610.04055 and is attached as an appendix. Theorem-numbering here matches the full version