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DOI: 10.4230/LIPIcs.FUN.2026.24

On the Complexity of the Maker-Breaker Happy Vertex Game

Authors: Mathieu Hilaire, Perig Montfort, and Nacim Oijid

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract

Given a c-colored graph G, a vertex v of G is said to be happy if it has the same color as all its neighbors. The notion of happy vertices was introduced by Zhang and Li [Peng Zhang and Angsheng Li, 2015] to compute the homophily of a graph. Eto, Fujimoto, Kiya, Matsushita, Miyano, Murao and Saitoh [Hiroshi Eto et al., 2025] introduced the Maker-Maker version of the Happy vertex game, where two players compete to claim more happy vertices than their opponent. We introduce here the Maker-Breaker happy vertex game: two players, Maker and Breaker, alternately color the vertices of a graph with their respective colors. Maker aims to maximize the number of happy vertices at the end, while Breaker aims to prevent her. This game is also a scoring version of the Maker-Breaker domination game introduced by Duchene, Gledel, Parreau and Renault [Duchene et al., 2020], as a happy vertex corresponds exactly to a vertex that is not dominated in the domination game. Therefore, this game is a very natural game on graphs and can be studied within the scope of scoring positional games [Bagan et al., 2024]. We initiate here the complexity study of this game, by proving that computing its score is PSPACE-complete on trees, NP-hard on caterpillars, and polynomial on subdivided stars. Finally, we provide the exact value of the score on graphs of maximum degree 2, and we provide an FPT-algorithm to compute the score on graphs of bounded neighborhood diversity. An important contribution of the paper is that, to achieve our hardness results, we introduce a new type of incidence graph called the literal-clause incidence graph for 2-SAT formulas. We prove that QMAX 2-SAT remains PSPACE-complete even if this graph is acyclic, and that MAX 2-SAT remains NP-complete, even if this graph is acyclic and has maximum degree 2, i.e. is a union of paths. We demonstrate the importance of this contribution by proving that Incidence, the scoring positional game played on a graph is also PSPACE-complete when restricted to forests.

Cite as

Mathieu Hilaire, Perig Montfort, and Nacim Oijid. On the Complexity of the Maker-Breaker Happy Vertex Game. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{hilaire_et_al:LIPIcs.FUN.2026.24,
  author =	{Hilaire, Mathieu and Montfort, Perig and Oijid, Nacim},
  title =	{{On the Complexity of the Maker-Breaker Happy Vertex Game}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.24},
  URN =		{urn:nbn:de:0030-drops-257434},
  doi =		{10.4230/LIPIcs.FUN.2026.24},
  annote =	{Keywords: Maker-Breaker game, Domination game, happy vertex game, scoring game, complexity}
}

Document

DOI: 10.4230/LIPIcs.STACS.2021.36

Reachability in Two-Parametric Timed Automata with One Parameter Is EXPSPACE-Complete

Authors: Stefan Göller and Mathieu Hilaire

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract

Parametric timed automata (PTA) have been introduced by Alur, Henzinger, and Vardi as an extension of timed automata in which clocks can be compared against parameters. The reachability problem asks for the existence of an assignment of the parameters to the non-negative integers such that reachability holds in the underlying timed automaton. The reachability problem for PTA is long known to be undecidable, already over three parametric clocks. A few years ago, Bundala and Ouaknine proved that for PTA over two parametric clocks and one parameter the reachability problem is decidable and also showed a lower bound for the complexity class PSPACE^NEXP. Our main result is that the reachability problem for parametric timed automata over two parametric clocks and one parameter is EXPSPACE-complete. For the EXPSPACE lower bound we make use of deep results from complexity theory, namely a serializability characterization of EXPSPACE (in turn based on Barrington’s Theorem) and a logspace translation of numbers in Chinese Remainder Representation to binary representation due to Chiu, Davida, and Litow. It is shown that with small PTA over two parametric clocks and one parameter one can simulate serializability computations. For the EXPSPACE upper bound, we first give a careful exponential time reduction from PTA over two parametric clocks and one parameter to a (slight subclass of) parametric one-counter automata over one parameter based on a minor adjustment of a construction due to Bundala and Ouaknine. For solving the reachability problem for parametric one-counter automata with one parameter, we provide a series of techniques to partition a fictitious run into several carefully chosen subruns that allow us to prove that it is sufficient to consider a parameter value of exponential magnitude only. This allows us to show a doubly-exponential upper bound on the value of the only parameter of a PTA over two parametric clocks and one parameter. We hope that extensions of our techniques lead to finally establishing decidability of the long-standing open problem of reachability in parametric timed automata with two parametric clocks (and arbitrarily many parameters) and, if decidability holds, determining its precise computational complexity.

Cite as

Stefan Göller and Mathieu Hilaire. Reachability in Two-Parametric Timed Automata with One Parameter Is EXPSPACE-Complete. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goller_et_al:LIPIcs.STACS.2021.36,
  author =	{G\"{o}ller, Stefan and Hilaire, Mathieu},
  title =	{{Reachability in Two-Parametric Timed Automata with One Parameter Is EXPSPACE-Complete}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{36:1--36:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.36},
  URN =		{urn:nbn:de:0030-drops-136817},
  doi =		{10.4230/LIPIcs.STACS.2021.36},
  annote =	{Keywords: Parametric Timed Automata, Computational Complexity, Reachability}
}

Document

DOI: 10.4230/LIPIcs.FUN.2016.10

Spy-Game on Graphs

Authors: Nathann Cohen, Mathieu Hilaire, Nícolas A. Martins, Nicolas Nisse, and Stéphane Pérennes

Published in: LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

Abstract

We define and study the following two-player game on a graph G. Let k in N^*. A set of k guards is occupying some vertices of G while one spy is standing at some node. At each turn, first the spy may move along at most s edges, where s in N^* is his speed. Then, each guard may move along one edge. The spy and the guards may occupy same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than d in N (a predefined distance) from every guard. Can the spy win against k guards? Similarly, what is the minimum distance d such that k guards may ensure that at least one of them remains at distance at most d from the spy? This game generalizes two well-studied games: Cops and robber games (when s=1) and Eternal Dominating Set (when s is unbounded). We consider the computational complexity of the problem, showing that it is NP-hard and that it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards and the required distance d when G is a path or a cycle. Our main result is that there exists beta>0 such that Omega(n^{1+beta}) guards are required to win in any n*n grid.

Cite as

Nathann Cohen, Mathieu Hilaire, Nícolas A. Martins, Nicolas Nisse, and Stéphane Pérennes. Spy-Game on Graphs. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cohen_et_al:LIPIcs.FUN.2016.10,
  author =	{Cohen, Nathann and Hilaire, Mathieu and Martins, N{\'\i}colas A. and Nisse, Nicolas and P\'{e}rennes, St\'{e}phane},
  title =	{{Spy-Game on Graphs}},
  booktitle =	{8th International Conference on Fun with Algorithms (FUN 2016)},
  pages =	{10:1--10:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-005-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{49},
  editor =	{Demaine, Erik D. and Grandoni, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.10},
  URN =		{urn:nbn:de:0030-drops-58782},
  doi =		{10.4230/LIPIcs.FUN.2016.10},
  annote =	{Keywords: graph, two-player games, cops and robber games, complexity}
}

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Documents authored by Hilaire, Mathieu

On the Complexity of the Maker-Breaker Happy Vertex Game

Abstract

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Reachability in Two-Parametric Timed Automata with One Parameter Is EXPSPACE-Complete

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Spy-Game on Graphs

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