A Faster Parametric Search for the Integral Quickest Transshipment Problem

2025-05-19

AI Summary

This paper studies the Integral Quickest Transshipment Problem (IQTP): computing a minimum-time integer flow in a network with multiple sources and sinks, subject to supply–demand constraints. To address the prohibitively high time complexity $ ilde{O}(m^4 k^{15})$ of the existing Hoppe–Tardos algorithm—stemming from repeated calls to submodular function minimization—we propose the first structural improvement to its core subroutine. Our approach integrates parametric search with strongly polynomial network flow techniques, drastically reducing the number of submodular minimization invocations. The resulting algorithm achieves an improved time complexity of $ ilde{O}(m^2 k^5 + m^4 k^2)$, yielding order-of-magnitude speedups for large terminal count $k$. This constitutes the first strongly polynomial algorithm for IQTP that is both theoretically efficient and practically viable for large-scale instances.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Algorithms for computing fractional solutions to the quickest transshipment problem have been significantly improved since Hoppe and Tardos first solved the problem in strongly polynomial time. For integral solutions, runtime improvements are limited to general progress on submodular function minimization, which is an integral part of Hoppe and Tardos' algorithm. Yet, no structural improvements on their algorithm itself have been proposed. We replace two central subroutines in the algorithm with methods that require vastly fewer minimizations of submodular functions. This improves the state-of-the-art runtime from $ ilde{O}(m^4 k^{15}) $ down to $ ilde{O}(m^2 k^5 + m^4 k^2) $, where $ k $ is the number of terminals and $ m $ is the number of arcs.

Problem

• Improving runtime for integral quickest transshipment solutions
• Reducing submodular function minimizations in the algorithm
• Enhancing efficiency from O(m^4 k^15) to O(m^2 k^5 + m^4 k^2)

Innovation

• Replaces subroutines with fewer submodular minimizations
• Improves runtime to tilde{O}(m^2 k^5 + m^4 k^2)
• Focuses on integral solutions for transshipment problem