[Paper Review] Maker-Breaker is solved in polynomial time on hypergraphs of rank 3

Florian Galliot, Sylvain Gravier+1 more|arXiv (Cornell University)|Sep 26, 2022

Artificial Intelligence in Games13 references4 citations

TL;DR

This paper introduces a danger-based structural framework to solve the Maker-Breaker game on 3-uniform hypergraphs in polynomial time. By characterizing Breaker wins via intersection properties of specific danger families, it proves that the game is decidable in polynomial time—resolving a conjecture by Rahman and Watson—and establishes that Maker can win in logarithmic rounds relative to the number of vertices.

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).

Motivation & Objective

• To resolve the algorithmic complexity of the Maker-Breaker game on hypergraphs of rank 3, which had been conjectured to be tractable.
• To develop a structural characterization of Breaker wins based on the intersection of defined danger subhypergraphs.
• To provide a polynomial-time decision algorithm for determining the winner of the Maker-Breaker game on 3-uniform hypergraphs.
• To establish tight upper bounds on the number of rounds required for Maker to win, showing logarithmic dependency on vertex count.
• To validate Rahman and Watson’s conjecture that the game is polynomial-time solvable on hypergraphs of rank 3.

Proposed method

• Introduces the concept of 'danger' at a vertex as a subhypergraph representing an urgent threat that Breaker must block if Maker claims that vertex.
• Defines a family of elementary dangers, denoted $\mathcal{F}$, such that a hypergraph is a Breaker win iff all $\mathcal{F}$-dangers intersect at every vertex.
• Applies the danger framework to 3-uniform marked hypergraphs, where vertices are marked as claimed by Maker.
• Uses structural analysis of $\mathcal{D}_1^{\text{O,rest}}$-dangers and their intersections to derive decision criteria.
• Constructs a polynomial-time algorithm by checking for the presence of specific elementary subhypergraphs (nunchaku or necklace) within the first three rounds.
• Employs preprocessing and redundant computation reduction to optimize the time complexity to $O(\max(n^5m^2, n^6\Delta))$.

Experimental results

Research questions

• RQ1 Can the Maker-Breaker game on hypergraphs of rank 3 be decided in polynomial time?
• RQ2 What structural condition on dangers at each vertex determines whether Breaker has a winning strategy?
• RQ3 Can Maker guarantee a win within a logarithmic number of rounds relative to the number of unclaimed vertices?
• RQ4 Does the absence of certain elementary subhypergraphs (nunchaku or necklace) in the first three rounds imply a Breaker win?
• RQ5 Does the danger-based framework generalize to the biased or 3-CNF formula variants of the game?

Key findings

• The Maker-Breaker game on hypergraphs of rank 3 is solvable in polynomial time, resolving a conjecture by Rahman and Watson.
• A hypergraph of rank 3 is a Breaker win if and only if all $\mathcal{F}$-dangers at every vertex intersect, where $\mathcal{F}$ is a constructed family of elementary dangers.
• Maker can win in at most $\lceil \log_2(|V(H)\setminus M(H)| - 5) \rceil$ rounds, establishing a nearly optimal logarithmic bound.
• The algorithmic decision procedure runs in $O(\max(n^5m^2, n^6\Delta))$ time, with optimized preprocessing.
• The result implies that Maker wins if and only if he can force the appearance of a nunchaku or necklace within the first three rounds.
• The paper provides a counterexample to a previously claimed similar result, ensuring the correctness of the current framework.