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On the Complexity of Client-Waiter and Waiter-Client Games

Authors Valentin Gledel , Nacim Oijid , Sébastien Tavenas , Stéphan Thomassé

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ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Mathematics of computing → Hypergraphs
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Complexity
  • positional games
  • Maker-Breaker
  • Client-Waiter
  • Waiter-Client
  • PSPACE-complete
  • FPT

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Abstract

Positional games were introduced by Hales and Jewett in 1963, and their study became more popular when Erdős and Selfridge showed their connection to Ramsey theory and hypergraph coloring in 1973. Several conventions of these games exist, and the most popular one, Maker-Breaker was proved to be PSPACE-complete by Schaefer in 1978. The study of their complexity then stopped for decades, until 2017 when Bonnet, Jamain, and Saffidine proved that Maker-Breaker is W[1]-complete when parameterized by the number of moves. The study was then intensified when Rahman and Watson improved Schaefer’s result in 2021 by proving that the PSPACE-hardness holds for 6-uniform hypergraphs. More recently, Galliot, Gravier, and Sivignon proved that computing the winner on rank 3 hypergraphs is in P, and Keopke proved that the PSPACE-hardness also holds for 5-uniform hypergraphs. 
We focus here on the Client-Waiter and the Waiter-Client conventions. Both were proved to be NP-hard by Csernenszky, Martin, and Pluhár in 2011, but neither completeness nor positive results were known. In this paper, we complete the study of these conventions by proving that the former is PSPACE-complete, even restricted to 6-uniform hypergraphs, and by providing an FPT-algorithm for the latter, parameterized by the size of its largest edge. In particular, the winner of Waiter-Client can be computed in polynomial time in rank k hypergraphs for any fixed integer k. Finally, in search of the exact location of the complexity gap in the Client-Waiter convention, we focus on rank 3 hypergraphs. We provide an algorithm that runs in polynomial time with an oracle in NP.

Cite As Get BibTex

Valentin Gledel, Nacim Oijid, Sébastien Tavenas, and Stéphan Thomassé. On the Complexity of Client-Waiter and Waiter-Client Games. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 89:1-89:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.89

Author Details

Valentin Gledel
  • Université Savoie Mont Blanc, CNRS UMR5127, LAMA, Chambéry, F-73000, France
Nacim Oijid
  • Lebanese American University, Univ Lyon, CNRS, INSA Lyon, UCBL, Centrale Lyon, Univ Lyon 2, LIRIS, UMR5205, F-69622 Villeurbanne, France
Sébastien Tavenas
  • Université Savoie Mont Blanc, CNRS UMR5127, LAMA, Chambéry, F-73000, France
Stéphan Thomassé
  • Univ Lyon, EnsL, UCBL, CNRS, LIP, F-69342, LYON Cedex 07, France

Funding

This research was partly supported by the ANR project P-GASE (ANR-21-CE48-0001-01).

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