Sequentially swapping tokens: further on graph classes

DOI10.1007/978-3-031-23101-8_15zbMath1529.68218arXiv2210.02835OpenAlexW4313429656MaRDI QIDQ6169527

Hirotaka Ono, Yuto Okada, Yota Otachi, Hironori Kiya

Publication date: 14 August 2023

Published in: Lecture Notes in Computer Science (Search for Journal in Brave)

Abstract: We study the following variant of the 15 puzzle. Given a graph and two token placements on the vertices, we want to find a walk of the minimum length (if any exists) such that the sequence of token swappings along the walk obtains one of the given token placements from the other one. This problem was introduced as Sequential Token Swapping by Yamanaka et al. [JGAA 2019], who showed that the problem is intractable in general but polynomial-time solvable for trees, complete graphs, and cycles. In this paper, we present a polynomial-time algorithm for block-cactus graphs, which include all previously known cases. We also present general tools for showing the hardness of the problem on restricted graph classes such as chordal graphs and chordal bipartite graphs. We also show that the problem is hard on grids and king's graphs, which are the graphs corresponding to the 15 puzzle and its variant with relaxed moves.

Full work available at URL: https://arxiv.org/abs/2210.02835

zbMATH Keywords

grid graphblock-cactus graphKing's graph(generalized) 15 puzzlesequential token swapping

Mathematics Subject Classification ID

Graph theory (including graph drawing) in computer science (68R10) Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) (68Q17) Graph algorithms (graph-theoretic aspects) (05C85)

Cites Work

• The \((n^ 2-1)\)-puzzle and related relocation problems
• The NP-completeness of Steiner tree and dominating set for chordal bipartite graphs
• A simple proof that the \((n^{2} - 1)\)-puzzle is hard
• HAMILTONian circuits in chordal bipartite graphs
• Introduction to reconfiguration
• Graph Theory
• The complexity of change
• Steiner trees, connected domination and strongly chordal graphs
• Fast Pattern Matching in Strings
• The Rectilinear Steiner Tree Problem is $NP$-Complete
• Sequentially Swapping Colored Tokens on Graphs
• Hamilton Paths in Grid Graphs

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