The time complexity of the token swapping problem and its parallel variants

Abstract

The token swapping problem (TSP) and its colored version are reconfiguration problems on graphs. This paper is concerned with the complexity of the TSP and two new variants; namely parallel TSP and parallel colored TSP. For a given graph where each vertex has a unique token on it, the TSP requires to find a shortest way to modify a token placement into another by swapping tokens on adjacent vertices. In the colored version, vertices and tokens are colored and the goal is to relocate tokens so that each vertex has a token of the same color. Their parallel versions allow simultaneous swaps on non-incident edges in one step. We investigate the time complexity of several restricted cases of those problems and show when those problems become tractable and remain intractable.