Computational Complexity of Generalized Push Fight

TL;DR

This paper investigates the computational complexity of Push Fight, proving it is PSPACE-hard in general, but identifying specific cases like mate-in-1 as polynomial-time solvable or NP-complete depending on move constraints.

Contribution

It establishes the PSPACE-hardness of Push Fight and characterizes the complexity of mate-in-1 problems under various move constraints.

Findings

Push Fight is PSPACE-hard in general.

Mate-in-1 is polynomial-time solvable with a fixed number of moves.

Mate-in-1 becomes NP-complete when the number of moves per turn is part of the input.

Abstract

We analyze the computational complexity of optimally playing the two-player board game Push Fight, generalized to an arbitrary board and number of pieces. We prove that the game is PSPACE-hard to decide who will win from a given position, even for simple (almost rectangular) hole-free boards. We also analyze the mate-in-1 problem: can the player win in a single turn? One turn in Push Fight consists of up to two "moves" followed by a mandatory "push". With these rules, or generalizing the number of allowed moves to any constant, we show mate-in-1 can be solved in polynomial time. If, however, the number of moves per turn is part of the input, the problem becomes NP-complete. On the other hand, without any limit on the number of moves per turn, the problem becomes polynomially solvable again.