A Separator Theorem for Hypergraphs and a CSP-SAT Algorithm

• Published in: arXiv.org
• Main Authors: Koucký, Michal, Rödl, Vojtěch, Talebanfard, Navid
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 10.12.2021
• Subjects: Algorithms; Computer Science - Computational Complexity; Computer Science - Logic in Computer Science; Graph theory; Separators
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2105.06744

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.