Settling Weighted Token Swapping up to Algorithmic Barriers

TL;DR

This paper introduces the first approximation algorithms and tight barrier results for the weighted token swapping problem on graphs and trees, extending the understanding from the unweighted case to weighted scenarios.

Contribution

It provides the first known approximation algorithms for weighted token swapping on graphs and trees, along with tight barrier results, addressing a significant gap in the literature.

Findings

Approximation ratio of 2+2W/w for general graphs.

Approximation ratio of 1+W/w for trees.

Established tight barrier results for weighted token swapping.

Abstract

We study the weighted token swapping problem, in which we are given a graph on $n$ vertices, $n$ weighted tokens, an initial assignment of one token to each vertex, and a final assignment of one token to each vertex. The goal is to find a minimum-cost sequence of swaps of adjacent tokens to reach the final assignment from the initial assignment, where the cost is the sum over all swaps of the sum of the weights of the two swapped tokens. Unweighted token swapping has been extensively studied: it is NP-hard to approximate to a factor better than $14/13$, and there is a polynomial-time 4-approximation, along with a tight "barrier" result showing that the class of locally optimal algorithms cannot achieve a ratio better than 4. For trees, the problem remains NP-hard to solve exactly, and there is a polynomial-time 2-approximation, along with a tight barrier result showing that the class of $\ell$-straying algorithms cannot achieve a ratio better than 2. Weighted token swapping with $\{0,1\}$ weights is much harder to approximation: it is NP-hard to approximate even to a factor of $(1-\varepsilon) \cdot \ln n$ for any constant $\varepsilon>0$. Restricting to positive weights, no approximation algorithms are known, and the only known lower bounds are those inherited directly from the unweighted version. We provide the first approximation algorithms for weighted token swapping on both trees and general graphs, along with tight barrier results. Letting $w$ and $W$ be the minimum and maximum token weights, our approximation ratio is $2+2W/w$ for general graphs and $1+W/w$ for trees.