Solving the General Case of Rank-3 Maker-Breaker Games in Polynomial Time

TL;DR

This paper proves that determining the winner in the general rank-3 Maker-Breaker game on hypergraphs can be solved efficiently in polynomial time, extending previous results from special cases.

Contribution

It establishes that the general case of rank-3 Maker-Breaker games is solvable in polynomial time, broadening the scope of efficiently solvable game configurations.

Findings

General rank-3 Maker-Breaker game is polynomial-time solvable.

Extends previous polynomial results from degenerate cases.

Provides a comprehensive solution method for the general case.

Abstract

A rank-3 Maker-Breaker game is played on a hypergraph in which all hyperedges are sets of at most 3 vertices. The two players of the game, called Maker and Breaker, move alternately. On his turn, maker chooses a vertex to be withdrawn from all hyperedges, while Breaker on her turn chooses a vertex and delete all the hyperedges containing that vertex. Maker wins when by the end of his turn some hyperedge is completely covered, i.e. the last remaining vertex of that hyperedge is withdrawn. Breaker wins when by the end of her turn, all hyperedges have been deleted. Solving a Maker-Breaker game is the computational problem of choosing an optimal move, or equivalently, deciding which player has a winning strategy in a configuration. The complexity of solving two degenerate cases of rank-3 games has been proven before to be polynomial. In this paper, we show that the general case of rank-3 Maker-Breaker games is also solvable in polynomial time.