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Erdős-Selfridge Theorem for Nonmonotone CNFs

Authors Md Lutfar Rahman, Thomas Watson



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Author Details

Md Lutfar Rahman
  • Amazon, Seattle, WA, USA
Thomas Watson
  • University of Memphis, TN, USA

Acknowledgements

We thank anonymous reviewers for their comments.

Cite AsGet BibTex

Md Lutfar Rahman and Thomas Watson. Erdős-Selfridge Theorem for Nonmonotone CNFs. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 31:1-31:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SWAT.2022.31

Abstract

In an influential paper, Erdős and Selfridge introduced the Maker-Breaker game played on a hypergraph, or equivalently, on a monotone CNF. The players take turns assigning values to variables of their choosing, and Breaker’s goal is to satisfy the CNF, while Maker’s goal is to falsify it. The Erdős-Selfridge Theorem says that the least number of clauses in any monotone CNF with k literals per clause where Maker has a winning strategy is Θ(2^k). We study the analogous question when the CNF is not necessarily monotone. We prove bounds of Θ(√2 ^k) when Maker plays last, and Ω(1.5^k) and O(r^k) when Breaker plays last, where r = (1+√5)/2≈ 1.618 is the golden ratio.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
Keywords
  • Game
  • nonmonotone
  • CNFs

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