Approximation and Hardness for Token Swapping

TL;DR

This paper studies the computational complexity and approximation algorithms for the token swapping problem on graphs, establishing tight bounds, hardness results, and extending to a colored version.

Contribution

It provides tight bounds for exact and approximation algorithms, proves APX-hardness, and extends results to a colored token swapping variant.

Findings

Exact algorithms cannot be subexponential under ETH.

A simple $2^{O(n ext{log} n)}$ algorithm exists.

A 4-approximation algorithm is achievable, with APX-hardness proven.

Abstract

Given a graph $G=(V,E)$ with $V=\{1,\ldots,n\}$, we place on every vertex a token $T_1,\ldots,T_n$. A swap is an exchange of tokens on adjacent vertices. We consider the algorithmic question of finding a shortest sequence of swaps such that token $T_i$ is on vertex $i$. We are able to achieve essentially matching upper and lower bounds, for exact algorithms and approximation algorithms. For exact algorithms, we rule out any $2^{o(n)}$ algorithm under the ETH. This is matched with a simple $2^{O(n\log n)}$ algorithm based on a breadth-first search in an auxiliary graph. We show one general $4$-approximation and show APX-hardness. Thus, there is a small constant $\delta>1$ such that every polynomial time approximation algorithm has approximation factor at least $\delta$. Our results also hold for a generalized version, where tokens and vertices are colored. In this generalized version each token must go to a vertex with the same color.