The Time Complexity of Permutation Routing via Matching, Token Swapping and a Variant

TL;DR

This paper investigates the computational complexity of permutation routing and token swapping problems on graphs, including a colored variant, identifying cases where these problems are tractable or intractable.

Contribution

It provides a detailed complexity analysis of various restricted cases of permutation routing and token swapping, including colored variants, highlighting when these problems are solvable efficiently.

Findings

Identifies tractable cases of permutation routing and token swapping.

Shows intractability results for certain restricted variants.

Analyzes the complexity of colored token swapping problems.

Abstract

The problems of Permutation Routing via Matching and Token Swapping are reconfiguration problems on graphs. This paper is concerned with the complexity of those problems and a colored variant. For a given graph where each vertex has a unique token on it, those problems require to find a shortest way to modify a token placement into another by swapping tokens on adjacent vertices. While all pairs of tokens on a matching can be exchanged at once in Permutation Routing via Matching, Token Swapping allows only one pair of tokens can be swapped. 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. We investigate the time complexity of several restricted cases of those problems and show when those problems become tractable and remain intractable.