PSPACE-completeness of k-Atropos

Abstract

Burke and Teng introduced a two-player combinatorial game Atropos based on Sperner’s lemma, and showed that deciding whether one has a winning strategy for Atropos is PSPACE-complete. In the original Atropos game, the players must color a node adjacent to the last colored node. Burke and Teng also mentioned a variant $k$-Atropos in which each move is at most of distance $k$ of the previous move, and asked a question on determining the computational complexity of this variant. In this paper, we answer this question by showing that for any fixed integer $k(k\ge 2)$, the associated decision problem $k$-Atropos is PSPACE-complete by reduction from True Quantified Boolean Formula (TQBF).