On the Complexity of Connection Games
This work addresses theoretical computer science problems for game complexity analysis, providing foundational results but is incremental in extending known complexity proofs to specific games.
The paper tackles the computational complexity of determining outcomes in three widely played connection games (Havannah, Twixt, and Slither), proving that this problem is PSPACE-complete for all three, and also analyzes parameterized complexities for generalizations of Hex.
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.