A Separator Theorem for Hypergraphs and a CSP-SAT Algorithm

TL;DR

This paper introduces a new separator theorem for hypergraphs with bounded degree, leading to an efficient algorithm for solving certain CSP-SAT problems and their refutations, advancing understanding of hypergraph structure and satisfiability.

Contribution

It presents a novel separator theorem for hypergraphs with bounded degree and applies it to develop faster algorithms for CSP-SAT, #CSP-SAT, and Max-CSP-SAT problems.

Findings

Separator theorem for hypergraphs with maximum degree o(√m)

Algorithm for CSP-SAT with time d^{(1-ε_r)m}

Refutation of unsatisfiable CSPs in size 2^{(1-ε_r)m}

Abstract

We show that for every $r \ge 2$ there exists $\epsilon_r > 0$ such that any $r$-uniform hypergraph with $m$ edges and maximum vertex degree $o(\sqrt{m})$ contains a set of at most $(\frac{1}{2} - \epsilon_r)m$ edges the removal of which breaks the hypergraph into connected components with at most $m/2$ edges. We use this to give an algorithm running in time $d^{(1 - \epsilon_r)m}$ that decides satisfiability of $m$-variable $(d, k)$-CSPs in which every variable appears in at most $r$ constraints, where $\epsilon_r$ depends only on $r$ and $k\in o(\sqrt{m})$. Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable $(2, k)$-CSPs with variable frequency $r$ can be refuted in tree-like resolution in size $2^{(1 - \epsilon_r)m}$. Furthermore for Tseitin formulas on graphs with degree at most $k$ (which are $(2, k)$-CSPs) we give a deterministic algorithm finding such a refutation.