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Documents authored by Oijid, Nacim

Document
DOI: 10.4230/LIPIcs.FUN.2026.5
Token Positional Games

Authors: Guillaume Bagan, Quentin Deschamps, Florian Galliot, Mirjana Mikalački, and Nacim Oijid

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

Abstract
The classical Maker-Breaker positional game is played on a board which is a hypergraph ℋ, with two players, Maker and Breaker, alternately claiming vertices of ℋ until all the vertices are claimed. When the game ends, Maker wins if she has claimed all the vertices of some edge of ℋ; otherwise, Breaker wins. Playing this game in real life can be done by placing tokens on the vertices of the board. In this paper, we study the unfortunate case in which one or both players do not have enough tokens to cover all the vertices and, as such, will have to move their tokens around at some point instead of placing new ones. There may be a bias, in that Maker and Breaker do not necessarily have the same amount of tokens. The present paper initiates the study of this generalization of positional games, called token positional games. A particularly interesting case is when Maker has a winning strategy in the classical game: what is the lowest number of tokens with which she still wins against Breaker’s unlimited stock? We notably show that, for k-uniform hypergraphs on an arbitrarily large number n of vertices, this number equals k if k ∈ {2,3} but can vary from k to Ω(n) if k ≥ 4. From an algorithmic point of view, PSPACE-hardness in general is inherited from classical positional games, but we get a polynomial-time algorithm to solve the case where Breaker only has one token. We also establish EXPTIME-completeness for a "token sliding" variation of the game.
Cite as

Guillaume Bagan, Quentin Deschamps, Florian Galliot, Mirjana Mikalački, and Nacim Oijid. Token Positional Games. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bagan_et_al:LIPIcs.FUN.2026.5,
  author =	{Bagan, Guillaume and Deschamps, Quentin and Galliot, Florian and Mikala\v{c}ki, Mirjana and Oijid, Nacim},
  title =	{{Token Positional Games}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{5:1--5:22},
  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.2026.5},
  URN =		{urn:nbn:de:0030-drops-257240},
  doi =		{10.4230/LIPIcs.FUN.2026.5},
  annote =	{Keywords: positional games, token games, hypergraphs, algorithmic complexity}
}
Document
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
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.89
On the Complexity of Client-Waiter and Waiter-Client Games

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

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

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

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)

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@InProceedings{gledel_et_al:LIPIcs.ICALP.2025.89,
  author =	{Gledel, Valentin and Oijid, Nacim and Tavenas, S\'{e}bastien and Thomass\'{e}, St\'{e}phan},
  title =	{{On the Complexity of Client-Waiter and Waiter-Client Games}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{89:1--89:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.89},
  URN =		{urn:nbn:de:0030-drops-234666},
  doi =		{10.4230/LIPIcs.ICALP.2025.89},
  annote =	{Keywords: Complexity, positional games, Maker-Breaker, Client-Waiter, Waiter-Client, PSPACE-complete, FPT}
}
Document
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.
Cite as

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.
Cite as

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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