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DOI: 10.4230/LIPIcs.ICALP.2017.90
URN: urn:nbn:de:0030-drops-74941
URL: http://drops.dagstuhl.de/opus/volltexte/2017/7494/
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Bonnet, Édouard ; Gaspers, Serge ; Lambilliotte, Antonin ; Rümmele, Stefan ; Saffidine, Abdallah

The Parameterized Complexity of Positional Games

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Abstract

We study the parameterized complexity of several positional games. Our main result is that Short Generalized Hex is W[1]-complete parameterized by the number of moves. This solves an open problem from Downey and Fellows’ influential list of open problems from 1999. Previously, the problem was thought of as a natural candidate for AW[*]-completeness. Our main tool is a new fragment of first-order logic where universally quantified variables only occur in inequalities. We show that model-checking on arbitrary relational structures for a formula in this fragment is W[1]-complete when parameterized by formula size. We also consider a general framework where a positional game is represented as a hypergraph and two players alternately pick vertices. In a Maker-Maker game, the first player to have picked all the vertices of some hyperedge wins the game. In a Maker-Breaker game, the first player wins if she picks all the vertices of some hyperedge, and the second player wins otherwise. In an Enforcer-Avoider game, the first player wins if the second player picks all the vertices of some hyperedge, and the second player wins otherwise. Short Maker-Maker, Short Maker-Breaker, and Short Enforcer-Avoider are respectively AW[*]-, W[1]-, and co-W[1]-complete parameterized by the number of moves. This suggests a rough parameterized complexity categorization into positional games that are complete for the first level of the W-hierarchy when the winning condition only depends on which vertices one player has been able to pick, but AW[*]-complete when it depends on which vertices both players have picked. However, some positional games with highly structured board and winning configurations are fixed-parameter tractable. We give another example of such a game, Short k-Connect, which is fixed-parameter tractable when parameterized by the number of moves.

BibTeX - Entry

@InProceedings{bonnet_et_al:LIPIcs:2017:7494,
  author =	{{\'E}douard Bonnet and Serge Gaspers and Antonin Lambilliotte and Stefan R{\"u}mmele and Abdallah Saffidine},
  title =	{{The Parameterized Complexity of Positional Games}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{90:1--90:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7494},
  URN =		{urn:nbn:de:0030-drops-74941},
  doi =		{10.4230/LIPIcs.ICALP.2017.90},
  annote =	{Keywords: Hex, Maker-Maker games, Maker-Breaker games, Enforcer-Avoider games, parameterized complexity theory}
}

Keywords: Hex, Maker-Maker games, Maker-Breaker games, Enforcer-Avoider games, parameterized complexity theory
Seminar: 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)
Issue Date: 2017
Date of publication: 06.07.2017


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