Playing Games with Algorithms: Algorithmic Combinatorial Game Theory

TL;DR

This paper surveys the field of combinatorial game theory, focusing on the complexity of solving perfect-information games and puzzles, highlighting open problems and recent algorithmic results.

Contribution

It provides an overview of combinatorial game theory and constraint logic, summarizing key complexity results and encouraging further research in the area.

Findings

Many combinatorial games are computationally hard to solve.

Some games have polynomial-time algorithms for ideal play.

The paper highlights open problems in the complexity of game solving.

Abstract

Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer.