arXiv subjects

Florian Galliot

Publications and source records attributed to Florian Galliot.

11 recordsLinked to original sources

arXiv2026

The disjoint separators problem in graphs

We study the disjoint separators problem in graphs, an analogue of the famous disjoint paths problem. Given a graph $G$ and four pairwise disjoint subsets of vertices $S_r$, $T_r$, $S_b$, $T_b$, we ask whether there exist an $(S_r,T_r)$-separator and an $(S_b,T_b)$-separator which are disjoint. This is equivalent to coloring the vertices in red or blue, with $S_r \cup T_r$ in red and $S_b \cup T_b$ in blue, such that there is no red $(S_r,T_r)$-path and no blue $(S_b,T_b)$-path. On the one hand, we show that the disjoint separators problem is NP-complete. We actually exhibit several NP-complete restrictions of this problem, including planar graphs of bounded maximum degree, and graphs of bounded maximum degree when $|S_r|=|T_r|=|S_b|=|T_b|=1$. On the other hand, these hardness results turn out to be quite tight, as we provide a structural characterization and a polynomial-time algorithm for planar graphs when $|S_r|=|T_r|=|S_b|=|T_b|=1$. This has an interesting consequence about the popular board game Hex: for the generalized game that may be played on any board, our result characterizes the planar boards on which draws are impossible, thus extending the well-known result about impossibility of draws on the standard commercialized board.

cs.DM↗

arXiv2026

A two-player version of the assignment problem

We introduce the competitive assignment problem, a two-player version of the well-known assignment problem. Given a set of tasks and a set of agents with different efficiencies for different tasks, Alice and Bob take turns picking agents one by one. Once all agents have been picked, Alice and Bob compute the optimal values $s_A$ and $s_B$ for the assignment problem on their respective sets of agents, i.e. they assign their own agents to tasks (with at most one agent per task and at most one task per agent) so as to maximize the sum of the efficiencies. The score of the game is then defined as $s_A-s_B$. Alice aims at maximizing the score, while Bob aims at minimizing it. This problem can model drafts in sports and card games, or more generally situations where two entities fight for the same resources and then use them to compete against each other. We show that the problem is PSPACE-complete, even restricted to agents that have at most two nonzero efficiencies. On the other hand, in the case of agents having at most one nonzero efficiency, the problem lies in XP parameterized by the number of tasks, and the optimal score can be computed in linear time when there are only two tasks.

cs.GT↗

arXiv2026

Token positional games

The classical Maker-Breaker positional game is played on a board which is a hypergraph $\mathcal{H}$, with two players, Maker and Breaker, alternately claiming vertices of $\mathcal{H}$ until all the vertices are claimed. When the game ends, Maker wins if she has claimed all the vertices of some edge of $\mathcal{H}$; 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 $k \in\{2,3\}$ but can vary from $k$ to $\Omega(n)$ if $k \geq 4$. 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.

cs.DM↗

arXiv2025

4-uniform Maker-Breaker and Maker-Maker games are PSPACE-complete

We study two positional games played on hypergraphs, whose edges may be interpreted as winning sets. Two players take turns picking a previously unpicked vertex of the hypergraph. We say a player fills an edge if that player has picked all the vertices of that edge. In the Maker-Maker convention, whoever first fills an edge wins, or we get a draw if no edge is filled. In the Maker-Breaker convention, the first player aims at filling an edge while the second player aims at preventing the first player from filling an edge. Our main result is that, for both games, deciding whether the first player has a winning strategy is a PSPACE-complete problem even when restricted to 4-uniform hypergraphs (of bounded maximum degree). For the Maker-Maker convention, this improves on the known PSPACE-completeness result for hypergraphs of rank 4. For the Maker-Breaker convention, this improves on the known PSPACE-completeness result for 5-uniform hypergraphs, and closes the complexity gap since the problem for hypergraphs of rank 3 is known to be solvable in polynomial time. As a corollary of our construction, we actually get a stronger result: deciding whether the first player has a winning strategy for the vertex-$C_4$-game played on arbitrary graphs, where the winning sets are the vertex sets of 4-cycles, is a PSPACE-complete problem for both conventions.

cs.DM↗

arXiv2025

Maker-Maker games of rank 4 are PSPACE-complete

The Maker-Maker convention of positional games is played on a hypergraph whose edges are interpreted as winning sets. Two players take turns picking a previously unpicked vertex, aiming at being first to pick all the vertices of some edge. Optimal play can only lead to a first player win or a draw, and deciding between the two is known to be PSPACE-complete even for 6-uniform hypergraphs. We establish PSPACE-completeness for hypergraphs of rank 4. As an intermediary, we use the recently introduced achievement positional games, a more general convention in which each player has their own winning sets (blue and red). We show that deciding whether the blue player has a winning strategy as the first player is PSPACE-complete even with blue edges of size 2 or 3 and pairwise disjoint red edges of size 2. The result for hypergraphs of rank 4 in the Maker-Maker convention follows as a simple corollary.

cs.DM↗

arXiv2025

The Avoider-Enforcer game on hypergraphs of rank 3

In the Avoider-Enforcer convention of positional games, two players, Avoider and Enforcer, take turns selecting vertices from a hypergraph H. Enforcer wins if, by the time all vertices of H have been selected, Avoider has completely filled an edge of H with her vertices; otherwise, Avoider wins. In this paper, we first give some general results, in particular regarding the outcome of the game and disjoint unions of hypergraphs. We then determine which player has a winning strategy for all hypergraphs of rank 2, and for linear hypergraphs of rank 3 when Avoider plays the last move. The structural characterisations we obtain yield polynomial-time algorithms.

math.CO↗

arXiv2025

A unified convention for achievement positional games

We introduce achievement positional games, a convention for positional games which encompasses the Maker-Maker and Maker-Breaker conventions. We consider two hypergraphs, one red and one blue, on the same vertex set. Two players, Left and Right, take turns picking a previously unpicked vertex. Whoever first fills an edge of their color, blue for Left or red for Right, wins the game (draws are possible). We establish general properties of such games. In particular, we show that a lot of principles which hold for Maker-Maker games generalize to achievement positional games. We also study the algorithmic complexity of deciding whether Left has a winning strategy as the first player when blue edges and red edges have respective sizes at most $p$ and $q$. This problem is in P for $p,q \leq 2$, but it is NP-hard for $p \geq 3$ and $q=2$, coNP-complete for $p=2$ and $q \geq 3$, and PSPACE-complete for $p,q \geq 3$ even when the 3-edges are the same for both colors. That last result has an interesting consequence on the Maker-Maker convention: for 3-uniform hypergraphs, which is the only case whose complexity is currently open (for starting positions of the game), we show PSPACE-completeness for positions obtained after one round of play.

cs.DM↗

arXiv2024

Poset Positional Games

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 analyse 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.

math.CO↗

arXiv2023

(k-2)-linear connected components in hypergraphs of rank k

We define a $q$-linear path in a hypergraph $H$ as a sequence $(e_1,\ldots,e_L)$ of edges of $H$ such that $|e_i \cap e_{i+1}| \in [\![1,q]\!]$ and $e_i \cap e_j=\varnothing$ if $|i-j|>1$. In this paper, we study the connected components associated to these paths when $q=k-2$ where $k$ is the rank of $H$. If $k=3$ then $q=1$ which coincides with the well-known notion of linear path or loose path. We describe the structure of the connected components, using an algorithmic proof which shows that the connected components can be computed in polynomial time. We then mention two consequences of our algorithmic result. The first one is that deciding the winner of the Maker-Breaker game on a hypergraph of rank 3 can be done in polynomial time. The second one is that tractable cases for the NP-complete problem of "Paths Avoiding Forbidden Pairs" in a graph can be deduced from the recognition of a special type of line graph of a hypergraph.

cs.DM↗

arXiv2023

An update on the coin-moving game on the square grid

This paper extends the work started in 2002 by Demaine, Demaine and Verill (DDV) on coin-moving puzzles. These puzzles have a long history in the recreational literature, but were first systematically analyzed by DDV, who gave a full characterization of the solvable puzzles on the triangular grid and a partial characterization of the solvable puzzles on the square grid. This article specifically extends the study of the game on the square grid. Notably, DDV's result on puzzles with two "extra coins" is shown to be overly broad: this paper provides counterexamples as well as a revised version of this theorem. A new method for solving puzzles with two extra coins is then presented, which covers some cases where the aforementioned theorem does not apply. Puzzles with just one extra coin seem even more complicated, and are only touched upon by DDV. This paper delves deeper, studying a class of such puzzles that may be considered equivalent to a game of "poking" coins. Within this class, some cases are considered that are amenable to analysis.

cs.DM↗

arXiv2022

Maker-Breaker is solved in polynomial time on hypergraphs of rank 3

In the Maker-Breaker positional game, Maker and Breaker take turns picking vertices of a hypergraph $H$, and Maker wins if and only if she possesses all the vertices of some edge of $H$. Deciding the outcome (i.e., which player has a winning strategy) is a PSPACE-complete problem even when restricted to 4-uniform hypergraphs [Galliot, 2025]. As for hypergraphs of rank 3, the linear subcase (i.e., any two distinct edges intersect on at most one vertex) has been solved by obtaining a structural characterization of the outcome and a polynomial-time algorithm to decide it [Kutz, 2005]. A conjecture [Rahman and Watson, 2020] implies that the same results can be obtained for general hypergraphs of rank 3, which we confirm in this paper. We provide a structural characterization of the outcome and a description of both players' optimal strategies, all based on intersections of some key subhypergraph collections. From this, we derive a polynomial-time algorithm, thus closing the complexity gap for Maker-Breaker games relative to the size of the edges. Another corollary of our structural result is that, if Maker has a winning strategy on a hypergraph of rank 3, then she can ensure to win the game within a number of rounds that is logarithmic in the number of vertices.

cs.DM↗