Complexity of Token Swapping and its Variants

TL;DR

This paper investigates the computational complexity of Token Swapping and its variants, establishing hardness results and fixed-parameter tractability in certain graph classes, and exploring complexity boundaries in restricted graph families.

Contribution

It proves Token Swapping is W[1]-hard parameterized by swap sequence length and shows Subset Token Swapping is FPT in nowhere-dense graphs, advancing understanding of these problems' complexities.

Findings

Token Swapping is W[1]-hard with respect to swap sequence length.

Subset Token Swapping is FPT in nowhere-dense graph classes.

Complexity varies across graph classes, with some cases polynomial and others NP-hard.

Abstract

In the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is $W[1]$-hard parameterized by the length $k$ of a shortest sequence of swaps. In fact, we prove that, for any computable function $f$, it cannot be solved in time $f(k)n^{o(k / \log k)}$ where $n$ is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial $n^{O(k)}$-time algorithm. We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have different colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases.