License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2022.31
URN: urn:nbn:de:0030-drops-161916
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16191/
Rahman, Md Lutfar ;
Watson, Thomas
Erdős-Selfridge Theorem for Nonmonotone CNFs
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.
BibTeX - Entry
@InProceedings{rahman_et_al:LIPIcs.SWAT.2022.31,
author = {Rahman, Md Lutfar and Watson, Thomas},
title = {{Erd\H{o}s-Selfridge Theorem for Nonmonotone CNFs}},
booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
pages = {31:1--31:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-236-5},
ISSN = {1868-8969},
year = {2022},
volume = {227},
editor = {Czumaj, Artur and Xin, Qin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16191},
URN = {urn:nbn:de:0030-drops-161916},
doi = {10.4230/LIPIcs.SWAT.2022.31},
annote = {Keywords: Game, nonmonotone, CNFs}
}
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Keywords: |
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Game, nonmonotone, CNFs |
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Collection: |
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18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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22.06.2022 |
