Florian Galliot

Florian Galliot

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.

Florian Galliot, Jonas Sénizergues

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.