A Faster Algorithm for Quickest Transshipments via an Extended Discrete Newton Method

TL;DR

This paper introduces a significantly faster algorithm for the Quickest Transshipment Problem, improving computational efficiency by extending the Discrete Newton Method, making solutions more practical for real-world applications.

Contribution

It presents a novel extension of the Discrete Newton Method that reduces the algorithm's running time from previous methods relying on parametric submodular function minimization.

Findings

Reduces running time from $ ilde{O}(m^4k^{14})$ to $ ilde O(m^2k^5+m^3k^3+m^3n)$

Provides a more practical approach for solving the Quickest Transshipment Problem

Enhances efficiency for large-scale network flow problems

Abstract

The Quickest Transshipment Problem is to route flow as quickly as possible from sources with supplies to sinks with demands in a network with capacities and transit times on the arcs. It is of fundamental importance for numerous applications in areas such as logistics, production, traffic, evacuation, and finance. More than 25 years ago, Hoppe and Tardos presented the first (strongly) polynomial-time algorithm for this problem. Their approach, as well as subsequently derived algorithms with strongly polynomial running time, are hardly practical as they rely on parametric submodular function minimization via Megiddo's method of parametric search. The main contribution of this paper is a considerably faster algorithm for the Quickest Transshipment Problem that instead employs a subtle extension of the Discrete Newton Method. This improves the previously best known running time of $\tilde{O}(m^4k^{14})$ to $\tilde O(m^2k^5+m^3k^3+m^3n)$, where $n$ is the number of nodes, $m$ the number of arcs, and $k$ the number of sources and sinks.