A Separator Theorem for Hypergraphs and a CSP-SAT Algorithm

• Published in: Logical methods in computer science 2021-01, Vol.17, Issue 4
• Main Authors: Koucký, Michal, Rödl, Vojtěch, Talebanfard, Navid
• Format: Article
• Language: English
• Subjects: computer science - computational complexity; computer science - logic in computer science
• ISSN: 1860-5974; 1860-5974
• DOI: 10.46298/lmcs-17(4:17)2021

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.