Journal Article Swapping Colored Tokens on Graphs

Yamanaka, Katsuhisa  ,  Horiyama, Takashi  ,  Kirkpatrick, David  ,  Otachi, Yota  ,  Saitoh, Toshiki  ,  Uehara, Ryuhei  ,  Uno, Yushi

9214pp.619 - 628 , 2015-08-05 , Springer
We investigate the computational complexity of the followingproblem. 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 as small numbers of swappings as possible 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.
Algorithms and Data Structures, 14th International Symposium, WADS 2015, Victoria, BC, Canada, August 5-7, 2015. Proceedings

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