Maker-Breaker is solved in polynomial time on hypergraphs of rank 3
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| Format: | Preprint |
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2022
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| _version_ | 1866916659719045120 |
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| author | Galliot, Florian Gravier, Sylvain Sivignon, Isabelle |
| author_facet | Galliot, Florian Gravier, Sylvain Sivignon, Isabelle |
| contents | In the Maker-Breaker positional game, Maker and Breaker take turns picking vertices of a hypergraph $H$, and Maker wins if and only if she possesses all the vertices of some edge of $H$. Deciding the outcome (i.e. which player has a winning strategy) is PSPACE-complete even when restricted to 5-uniform hypergraphs (Koepke, 2025). On hypergraphs of rank 3, a structural characterization of the outcome and a polynomial-time algorithm have been obtained for two subcases: one by Kutz (2005), the other by Rahman and Watson (2020) who conjectured that their result should generalize to all hypergraphs of rank 3. We prove this conjecture through a structural characterization of the outcome and a description of both players' optimal strategies, all based on intersections of some key subhypergraph collections, from which we derive a polynomial-time algorithm. Another corollary of our structural result is that, if Maker has a winning strategy on a hypergraph of rank 3, then she can ensure to win the game in a number of rounds that is logarithmic in the number of vertices. Note: This paper provides a counterexample to a similar result which was incorrectly claimed (arXiv:2209.11202, Theorem 22). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_12819 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Maker-Breaker is solved in polynomial time on hypergraphs of rank 3 Galliot, Florian Gravier, Sylvain Sivignon, Isabelle Discrete Mathematics Combinatorics In the Maker-Breaker positional game, Maker and Breaker take turns picking vertices of a hypergraph $H$, and Maker wins if and only if she possesses all the vertices of some edge of $H$. Deciding the outcome (i.e. which player has a winning strategy) is PSPACE-complete even when restricted to 5-uniform hypergraphs (Koepke, 2025). On hypergraphs of rank 3, a structural characterization of the outcome and a polynomial-time algorithm have been obtained for two subcases: one by Kutz (2005), the other by Rahman and Watson (2020) who conjectured that their result should generalize to all hypergraphs of rank 3. We prove this conjecture through a structural characterization of the outcome and a description of both players' optimal strategies, all based on intersections of some key subhypergraph collections, from which we derive a polynomial-time algorithm. Another corollary of our structural result is that, if Maker has a winning strategy on a hypergraph of rank 3, then she can ensure to win the game in a number of rounds that is logarithmic in the number of vertices. Note: This paper provides a counterexample to a similar result which was incorrectly claimed (arXiv:2209.11202, Theorem 22). |
| title | Maker-Breaker is solved in polynomial time on hypergraphs of rank 3 |
| topic | Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2209.12819 |