### References & Citations

# Computer Science > Logic in Computer Science

# Title: A Separator Theorem for Hypergraphs and a CSP-SAT Algorithm

(Submitted on 14 May 2021 (v1), revised 22 Sep 2021 (this version, v2),

*latest version 17 Oct 2021*(v3))Abstract: We show that for every $r \ge 2$ there exists $\epsilon_r > 0$ such that any $r$-uniform hypergraph on $m$ edges with bounded vertex degree has 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.

## Submission history

From: Navid Talebanfard [view email]**[v1]**Fri, 14 May 2021 10:09:18 GMT (16kb)

**[v2]**Wed, 22 Sep 2021 13:40:21 GMT (17kb)

**[v3]**Sun, 17 Oct 2021 21:14:51 GMT (21kb)

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