On the Complexity of Connection Games

TL;DR

This paper proves that determining the outcome of three popular connection games is computationally complex, specifically PSPACE-complete, and explores their parameterized complexity and solution limitations.

Contribution

It establishes the PSPACE-completeness of Havannah, Twixt, and Slither, and analyzes the parameterized complexity of generalized Hex variants.

Findings

Outcome determination is PSPACE-complete for all three games.

Short Generalized Hex is W[1]-hard, but Short Hex is FPT.

The ultra-weak solution for Hex cannot be fully adapted to these games.

Abstract

In this paper, we study three connection games among the most widely played: Havannah, Twixt, and Slither. We show that determining the outcome of an arbitrary input position is PSPACE-complete in all three cases. Our reductions are based on the popular graph problem Generalized Geography and on Hex itself. We also consider the complexity of generalizations of Hex parameterized by the length of the solution and establish that while Short Generalized Hex is W[1]-hard, Short Hex is FPT. Finally, we prove that the ultra-weak solution to the empty starting position in hex cannot be fully adapted to any of these three games.