Abstract
We consider a puzzle consisting of colored tokens on an n-vertex graph, where each token has a distinct starting vertex and a set of allowable target vertices for it to reach, and the only allowed transformation is to “sequentially” move the chosen token along a path of the graph by swapping it with other tokens on the path. This puzzle is a variation of the Fifteen Puzzle and is solvable in \(\text{ O }(n^3)\) token-swappings. We thus focus on the problem of minimizing the number of token-swappings to reach the target token-placement. We first give an inapproximability result of this problem, and then show polynomial-time algorithms on trees, complete graphs, and cycles.
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- 1.
In this paper, we denote a walk of a graph by a sequence of vertices.
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Acknowledgment
This work is partially supported by MEXT/JSPS KAKENHI Grant Numbers JP24106002, JP24106004, JP24106005, JP24106007, JP24220003, JP24700008, JP26330004, JP26330009, JP26730001, JP15K00008, JP15K00009, JP16K00002, and JP16K16006, the Asahi Glass Foundation, JST, CREST, Foundations of Innovative Algorithms for Big Data, and JST, CREST, Foundations of Data Particlization for Next Generation Data Mining.
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Yamanaka, K. et al. (2017). Sequentially Swapping Colored Tokens on Graphs. In: Poon, SH., Rahman, M., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2017. Lecture Notes in Computer Science(), vol 10167. Springer, Cham. https://doi.org/10.1007/978-3-319-53925-6_34
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DOI: https://doi.org/10.1007/978-3-319-53925-6_34
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