Improved Hardness-of-Approximation for Token Swapping

TL;DR

This paper advances the understanding of the token swapping problem by establishing tighter hardness bounds for approximation ratios, especially for the weighted variant, and discusses limitations of existing algorithms.

Contribution

It improves the hardness of approximation lower bound for token swapping from 1001/1000 to 14/13 and analyzes the inapproximability of the weighted version, also highlighting barriers for current algorithms.

Findings

NP-hard to approximate token swapping within ratio 14/13

0/1-weighted token swapping is NP-hard to approximate within (1-ε)ln(n)

Current algorithms cannot surpass the 4-approximation ratio for standard token swapping

Abstract

We study the token swapping problem, in which we are given a graph with an initial assignment of one distinct token to each vertex, and a final desired assignment (again with one token per vertex). The goal is to find the minimum length sequence of swaps of adjacent tokens required to get from the initial to final assignment. The token swapping problem is known to be NP-complete. It is also known to have a polynomial-time 4-approximation algorithm. From the hardness-of-approximation side, it is known to be NP-hard to approximate with ratio better than 1001/1000. Our main result is an improvement of the approximation ratio of the lower bound: We show that it is NP-hard to approximate with ratio better than 14/13. We then turn our attention to the 0/1-weighted version, in which every token has a weight of either 0 or 1, and the cost of a swap is the sum of the weights of the two participating tokens. Unlike standard token swapping, no constant-factor approximation is known for this version, and we provide an explanation. We prove that 0/1-weighted token swapping is NP-hard to approximate with ratio better than $(1-\varepsilon) \ln(n)$ for any constant $\epsilon>0$. Lastly, we prove two barrier results for the standard (unweighted) token swapping problem. We show that one cannot beat the current best known approximation ratio of 4 using a large class of algorithms which includes all known algorithms, nor can one beat it using a common analysis framework.