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

TL;DR

This work completes the complexity classification by proving that 2-Solo Chess is NP-complete if the instance contains only knights or only kings.

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.

Figures (16)

• Figure 11: The WIRE in Knight 2-Solo Chess. Originally, all pieces have a budget of 2. We show that the knights drawn in red are virtual $0$-pieces.
• Figure 12: Two Knight Wires propagating a signal each.
• Figure 13: Two WIRE placements. Left: A wire turning a corner. Right: A standard wire turning into a thick wire.
• Figure 14: A wire crossing of a yellow and a blue wire utilizing the thick wire placement. Note that no yellow piece can capture any blue pieces and vice versa.
• Figure 15: Shifting a wire diagonally backward and forward respectively.

Theorems & Definitions (15)

• Lemma 10: Wire Lemma
• Lemma 11
• Corollary 12
• Lemma 13
• Lemma 14
• Lemma 15: 3-VAL Lemma
• Lemma 16: Decrement Lemma
• Lemma 17
• Lemma 18
• Lemma 18