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

TL;DR

This work resolves Maker-Breaker on 3-uniform hypergraphs by introducing a danger-intersection framework. It provides a complete structural characterization: Breaker wins exactly when a carefully crafted danger family $\mathcal{D}_2$ has no fork, and Maker wins otherwise by forcing a nunchaku or necklace within at most three rounds. The result yields a polynomial-time algorithm for deciding the outcome on rank-3 hypergraphs and shows Maker can finish a winning edge in $O(\log|V|)$ rounds when possible. The findings confirm Rahman and Watson’s conjecture for positive 3-CNF formulas and advance understanding of the complexity landscape for positional games, while also offering a tight bound on the duration of Maker's winning strategy.

Abstract

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 PSPACE-complete even when restricted to 5-uniform hypergraphs (Koepke, 2025). On hypergraphs of rank 3, a structural characterization of the outcome and a polynomial-time algorithm have been obtained for two subcases: one by Kutz (2005), the other by Rahman and Watson (2020) who conjectured that their result should generalize to all hypergraphs of rank 3. We prove this conjecture through a structural characterization of the outcome and a description of both players' optimal strategies, all based on intersections of some key subhypergraph collections, from which we derive a polynomial-time algorithm. 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 in a number of rounds that is logarithmic in the number of vertices. Note: This paper provides a counterexample to a similar result which was incorrectly claimed (arXiv:2209.11202, Theorem 22).

Figures (32)

• Figure 1: Evolution of the marked hypergraph during a game. The marked vertices are circled, as they will be in all figures.
• Figure 2: An $ab$-chain $P$ of length 0 (left), length 1 (middle), length 5 (right). In all figures of this paper, edges of size 3 will be represented using a "claw" shape joining their three vertices.
• Figure 3: An $a$-cycle $C$ of length 2 (left), length 3 (middle), length 5 (right). The outer vertices are highlighted, the others are inner vertices.
• Figure 4: An $a$-tadpole $T$ (that is not an $a$-cycle), two examples.
• Figure 5: From left to right: an $ab$-snake, an $ab$-nunchaku, an $a$-necklace.

Theorems & Definitions (178)

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