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Atropos-k is PSPACE-complete

Chao Yang, Zhujun Zhang

TL;DR

This work resolves the computational complexity of Atropos-k for all fixed k≥2, proving PSPACE-completeness via a reduction from True Quantified Boolean Formula (TQBF). The reduction embeds a quantified Boolean evaluation into a structured Atropos state using a rich set of gadgets (start, switch, multi-switch, merge, multi-merge, check, crossover) connected by path gadgets, first for Atropos-$2$ and then generalized to larger k by adjusting distances. The two-phase gameplay mirrors the quantifier alternation and clause satisfaction, ensuring a winning strategy exists exactly when the quantified formula is true. The results place Atropos-k among PSPACE-complete problems for all fixed k and outline parity-preserving gadget modifications; the infinite-k variant remains an open question.

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 Atropos-k 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>=2), Atropos-k is PSPACE-complete by reduction from True Quantified Boolean Formula (TQBF).

Atropos-k is PSPACE-complete

TL;DR

This work resolves the computational complexity of Atropos-k for all fixed k≥2, proving PSPACE-completeness via a reduction from True Quantified Boolean Formula (TQBF). The reduction embeds a quantified Boolean evaluation into a structured Atropos state using a rich set of gadgets (start, switch, multi-switch, merge, multi-merge, check, crossover) connected by path gadgets, first for Atropos- and then generalized to larger k by adjusting distances. The two-phase gameplay mirrors the quantifier alternation and clause satisfaction, ensuring a winning strategy exists exactly when the quantified formula is true. The results place Atropos-k among PSPACE-complete problems for all fixed k and outline parity-preserving gadget modifications; the infinite-k variant remains an open question.

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 Atropos-k 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>=2), Atropos-k is PSPACE-complete by reduction from True Quantified Boolean Formula (TQBF).
Paper Structure (3 sections, 2 theorems, 2 figures)

This paper contains 3 sections, 2 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Main Result
  3. Conclusion

Key Result

Theorem 1

Atropos is PSPACE-complete.

Figures (2)

  • Figure 1: An Atropos game board of size 7.
  • Figure 2: The overall structure.

Theorems & Definitions (2)

  • Theorem 1: bt08
  • Theorem 2