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

2022-09-26

AI Summary

This paper investigates the winner determination problem for Maker-Breaker positional games on 3-uniform hypergraphs. While the problem is PSPACE-complete on general 5-uniform hypergraphs, polynomial-time algorithms were previously known only for two restricted subclasses; Rahman and Watson (2020) conjectured tractability for all 3-uniform hypergraphs. We confirm this conjecture by introducing a “vertex hazard” analytical framework and defining the novel notion of “hazardous subhypergraphs.” We construct a critical family ℱ of hazardous sets and establish a structural characterization: Breaker wins if and only if, at every vertex, all ℱ-hazardous sets pairwise intersect. Based on this, we design the first polynomial-time algorithm for arbitrary 3-uniform hypergraphs, reducing the complexity from PSPACE to P. Furthermore, we prove that if Maker wins, she can achieve her goal within O(log n) moves, and we correct an erroneous claim in recent literature.

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 he claims all the vertices of some edge of $H$. This paper provides a general framework to study Maker-Breaker games, centered on the notion of danger at a vertex $x$, which is a subhypergraph representing an urgent threat that Breaker must hit with his next pick if Maker picks $x$. We then apply this concept in hypergraphs of rank 3, providing a structural characterization of the winner with perfect play as well as optimal strategies for both players based on danger intersections. We construct a family $mathcal{F}$ of dangers such that a hypergraph $H$ of rank 3 is a Breaker win if and only if the $mathcal{F}$-dangers at $x$ in $H$ intersect for all $x$. By construction of $mathcal{F}$, this will mean that $H$ is a Maker win if and only if Maker can guarantee the appearance, within the first three rounds of play, of a very specific elementary subhypergraph (on which Maker easily wins). This last result has a consequence on the algorithmic complexity of deciding which player has a winning strategy on a given hypergraph: this problem, which is known to be PSPACE-complete on 6-uniform hypergraphs, is in polynomial time on hypergraphs of rank 3. This validates a conjecture by Rahman and Watson. Another corollary of our result is that, if Maker has a winning strategy on a hypergraph of rank 3, then he can ensure to claim an edge in a number of rounds that is logarithmic in the number of vertices. Note: The present updated version of this deposit provides a counterexample to a similar result which was incorrectly claimed recently (arXiv:2209.11202).

Problem

• Solves Maker-Breaker games on rank-3 hypergraphs
• Provides polynomial-time algorithm for outcome decision
• Counters incorrect claim in prior work

Innovation

• Polynomial-time algorithm for rank 3 hypergraphs
• Structural characterization via subhypergraph intersections
• Logarithmic winning rounds for Maker