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DOI: 10.4230/LIPIcs.FUN.2024.2

Poset Positional Games

Authors: Guillaume Bagan, Eric Duchêne, Florian Galliot, Valentin Gledel, Mirjana Mikalački, Nacim Oijid, Aline Parreau, and Miloš Stojaković

Published in: LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

Abstract

We propose a generalization of positional games, supplementing them with a restriction on the order in which the elements of the board are allowed to be claimed. We introduce poset positional games, which are positional games with an additional structure - a poset on the elements of the board. Throughout the game play, based on this poset and the set of the board elements that are claimed up to that point, we reduce the set of available moves for the player whose turn it is - an element of the board can only be claimed if all the smaller elements in the poset are already claimed. We proceed to analyze these games in more detail, with a prime focus on the most studied convention, the Maker-Breaker games. First we build a general framework around poset positional games. Then, we perform a comprehensive study of the complexity of determining the game outcome, conditioned on the structure of the family of winning sets on the one side and the structure of the poset on the other.

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Guillaume Bagan, Eric Duchêne, Florian Galliot, Valentin Gledel, Mirjana Mikalački, Nacim Oijid, Aline Parreau, and Miloš Stojaković. Poset Positional Games. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bagan_et_al:LIPIcs.FUN.2024.2,
  author =	{Bagan, Guillaume and Duch\^{e}ne, Eric and Galliot, Florian and Gledel, Valentin and Mikala\v{c}ki, Mirjana and Oijid, Nacim and Parreau, Aline and Stojakovi\'{c}, Milo\v{s}},
  title =	{{Poset Positional Games}},
  booktitle =	{12th International Conference on Fun with Algorithms (FUN 2024)},
  pages =	{2:1--2:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-314-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{291},
  editor =	{Broder, Andrei Z. and Tamir, Tami},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.2},
  URN =		{urn:nbn:de:0030-drops-199100},
  doi =		{10.4230/LIPIcs.FUN.2024.2},
  annote =	{Keywords: Positional games, Maker-Breaker games, Game complexity, Poset, Connect 4}
}

Document

DOI: 10.4230/LIPIcs.STACS.2023.34

Avoidance Games Are PSPACE-Complete

Authors: Valentin Gledel and Nacim Oijid

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract

Avoidance games are games in which two players claim vertices of a hypergraph and try to avoid some structures. These games have been studied since the introduction of the game of SIM in 1968, but only few complexity results have been found out about them. In 2001, Slany proved some partial results on Avoider-Avoider games complexity, and in 2017 Bonnet et al. proved that short Avoider-Enforcer games are Co-W[1]-hard. More recently, in 2022, Miltzow and Stojaković proved that these games are NP-hard. As these games correspond to the misère version of the well-known Maker-Breaker games, introduced in 1963 and proven PSPACE-complete in 1978, one could expect these games to be PSPACE-complete too, but the question has remained open since then. Here, we prove here that both Avoider-Avoider and Avoider-Enforcer conventions are PSPACE-complete. Using the PSPACE-hardness of Avoider-Enforcer, we provide in appendix proofs that some particular Avoider-Enforcer games also are.

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Valentin Gledel and Nacim Oijid. Avoidance Games Are PSPACE-Complete. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gledel_et_al:LIPIcs.STACS.2023.34,
  author =	{Gledel, Valentin and Oijid, Nacim},
  title =	{{Avoidance Games Are PSPACE-Complete}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{34:1--34:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.34},
  URN =		{urn:nbn:de:0030-drops-176869},
  doi =		{10.4230/LIPIcs.STACS.2023.34},
  annote =	{Keywords: Games, Avoider-Enforcer, Maker-Breaker, Complexity, Avoider-Avoider, PSPACE-complete}
}

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Poset Positional Games

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Avoidance Games Are PSPACE-Complete

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