Swapping Colored Tokens on Graphs

TL;DR

This paper studies the complexity of token swapping on graphs with colored tokens, proving NP-completeness for three colors and polynomial solutions for two, with various graph restrictions and parameterized cases.

Contribution

It establishes NP-completeness for 3-colored token swapping on specific graphs and provides polynomial algorithms for 2-colored cases, advancing understanding of the problem's computational boundaries.

Findings

NP-complete for 3 colors on planar bipartite graphs

Polynomial-time solution for 2 colors on general graphs

Fixed-parameter tractability for complete graphs with respect to number of colors

Abstract

We investigate the computational complexity of the following problem. We are given a graph in which each vertex has an initial and a target color. 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 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. Besides, we show that, the problem for complete graphs is fixed-parameter tractable when parameterized by the number of colors, while it is known to be NP-complete when the number of colors is unbounded.