Computational Complexity of Jumping Block Puzzles

TL;DR

This paper explores the computational complexity of jumping block puzzles, a new variant inspired by combinatorial reconfiguration, and classifies several variants within complexity classes such as P, NP, and PSPACE.

Contribution

It introduces and analyzes the complexity of jumping block puzzles, extending the understanding of puzzle complexity classes beyond traditional sliding puzzles.

Findings

Certain variants are shown to be NP-complete.

Some variants are solvable in polynomial time.

Complexity classifications include P, NP, and PSPACE.

Abstract

In combinatorial reconfiguration, the reconfiguration problems on a vertex subset (e.g., an independent set) are well investigated. In these problems, some tokens are placed on a subset of vertices of the graph, and there are three natural reconfiguration rules called ``token sliding,'' ``token jumping,'' and ``token addition and removal''. In the context of computational complexity of puzzles, the sliding block puzzles play an important role. Depending on the rules and set of pieces, the sliding block puzzles characterize the computational complexity classes including P, NP, and PSPACE. The sliding block puzzles correspond to the token sliding model in the context of combinatorial reconfiguration. On the other hand, a relatively new notion of jumping block puzzles is proposed in puzzle society. This is the counterpart to the token jumping model of the combinatorial reconfiguration problems in the context of block puzzles. We investigate several variants of jumping block puzzles and determine their computational complexities.