We investigate the computational complexity of the following problem. We are given a graph in which each vertex has the current and target colors. Each pair of adjacent vertices can swap their current colors. Our goal is to perform the minimum number of swaps so that the current and target colors agree at each vertex. When the colors are chosen from {1, 2,..., c}, we call this problem c-COLORED TOKEN SWAPPING since the current color of a vertex can be seen as a colored token placed on the vertex. We show that c-COLORED TOKEN SWAPPING is NP-complete for every constant c ≥ 3 even if input graphs are restricted to connected planar bipartite graphs of maximum degree 3. We then show that 2-COLORED TOKEN SWAPPING can be solved in polynomial time for general graphs and in linear time for trees.
CITATION STYLE
Yamanaka, K., Horiyama, T., Kirkpatrick, D., Otachi, Y., Saitoh, T., Uehara, R., & Uno, Y. (2015). Swapping colored tokens on graphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9214, pp. 619–628). Springer Verlag. https://doi.org/10.1007/978-3-319-21840-3_51
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