Solving Maker-Breaker Games on 5-uniform hypergraphs is PSPACE-complete

TL;DR

This paper proves that determining the winner in Maker-Breaker games on 5-uniform hypergraphs is PSPACE-complete, extending previous results from 6-uniform hypergraphs and using a polynomial reduction from generalized geography.

Contribution

The paper establishes PSPACE-completeness for Maker-Breaker games on 5-uniform hypergraphs, improving prior NL-hardness results and providing a new polynomial reduction from generalized geography.

Findings

Deciding the winner is PSPACE-complete for 5-uniform hypergraphs.

Reduces generalized geography to Maker-Breaker game decision problem.

Extends complexity classification from 6-uniform to 5-uniform hypergraphs.

Abstract

Let $(X, \mathcal{F})$ be a hypergraph. The Maker-Breaker game on $(X, \mathcal{F})$ is a combinatorial game between two players, Maker and Breaker. Beginning with Maker, the players take turns claiming vertices from $X$ that have not yet been claimed. Maker wins if she manages to claim all vertices of some hyperedge $F \in \mathcal{F}$. Breaker wins if he claims at least one vertex in every hyperedge. M. L. Rahman and Thomas Watson proved in 2021 that, even when only Maker-Breaker games on 6-uniform hypergraphs are considered, the decision problem of determining which player has a winning strategy is PSPACE-complete. They also showed that the problem is NL-hard when considering hypergraphs of rank 5. In this paper, we improve the latter result by showing that deciding who wins Maker-Breaker games on 5-uniform hypergraphs is still a PSPACE-complete problem. We achieve this by polynomial transformation from the problem of solving the generalized geography game on bipartite digraphs with vertex degrees 3 or less, which is known to be PSPACE-complete.