Completing the Complexity Classification of 2-Solo Chess: Knights and Kings are Hard

TL;DR

This paper completes the complexity classification of 2-Solo Chess, proving NP-completeness for instances with only knights or only kings, thus expanding the understanding of its computational difficulty.

Contribution

It establishes that 2-Solo Chess is NP-complete for knight-only and king-only configurations, filling gaps in the previous complexity classification.

Findings

2-Solo Chess with only knights is NP-complete.

2-Solo Chess with only kings is NP-complete.

The problem is polynomial-time solvable for pawns, but NP-complete for rooks, bishops, and queens.

Abstract

We extend the study of the 2-Solo Chess problem which was first introduced by Aravind, Misra, and Mittal in 2022. 2-Solo Chess is a single-player variant of chess in which the player must clear the board via captures such that only one piece remains, with each piece capturing at most twice. It is known that the problem is solvable in polynomial time for instances containing only pawns, while it becomes NP-complete for instances restricted to rooks, bishops, or queens. In this work, we complete the complexity classification by proving that 2-Solo Chess is NP-complete if the instance contains only knights or only kings.