Settling Weighted Token Swapping up to Algorithmic Barriers

2025-07-30

AI Summary

This paper studies the weighted token swapping problem: given a graph, $n$ tokens with positive weights, and initial/target assignments, find a minimum-cost sequence of adjacent swaps, where each swap incurs cost equal to the sum of the weights of the two involved tokens. We present the first approximation algorithms for this NP-hard problem with provable guarantees. On general graphs, we achieve an approximation ratio of $2 + 2W/w$, where $W$ and $w$ denote the maximum and minimum token weights; on trees, the ratio improves to $1 + W/w$. Both bounds are tight, surpassing prior algorithms restricted to the unweighted case. Our approach integrates combinatorial optimization, local search, and weight-ratio analysis—yielding the first theoretically justified performance guarantees for the weighted setting. This work establishes foundational benchmark results for approximation algorithms in weighted token swapping.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Study weighted token swapping problem on graphs
• Find minimum-cost sequence of adjacent swaps
• Provide first approximation algorithms for trees and graphs

Innovation

• Approximation algorithm for weighted token swapping
• Tight barrier results for optimization limits
• Ratio based on min and max token weights