Fast Primal-Dual Update against Local Weight Update in Linear Assignment Problem and Its Application

TL;DR

This paper introduces a fast primal-dual update method for the dynamic weighted bipartite matching problem, enabling efficient maintenance of optimal matchings with local weight updates, and applies it to improve envy-cycle algorithms.

Contribution

It presents a novel primal-dual update approach that efficiently handles local weight updates in bipartite matching, reducing computational complexity for dynamic problems.

Findings

Single Dijkstra execution suffices for local updates

Algorithm achieves $ ext{O}(mn^2)$ time complexity

Improves envy-cycle procedure efficiency

Abstract

We consider a dynamic situation in the weighted bipartite matching problem: edge weights in the input graph are repeatedly updated and we are asked to maintain an optimal matching at any moment. A trivial approach is to compute an optimal matching from scratch each time an update occurs. In this paper, we show that if each update occurs locally around a single vertex, then a single execution of Dijkstra's algorithm is sufficient to preserve optimality with the aid of a dual solution. As an application of our result, we provide a faster implementation of the envy-cycle procedure for finding an envy-free allocation of indivisible items. Our algorithm runs in $\mathrm{O}(mn^2)$ time, while the known bound of the original one is $\mathrm{O}(mn^3)$, where $n$ and $m$ denote the numbers of agents and items, respectively.