Faster feasibility for dynamic flows and transshipments on temporal networks

2024-11-07

arXiv.org

AI Summary

This paper addresses the feasibility problem of dynamic transshipment in time-expanded networks with multiple sources and sinks: given time-varying edge capacities and traversal times, and node-wise time-dependent supply/demand constraints, determine whether a feasible flow satisfying all supplies and demands exists. We propose the first strongly polynomial-time exact algorithm. Our approach combines time-expanded graph cut analysis, the Hoppe–Tardos reduction, and piecewise-constant function modeling to reformulate the problem into a canonical form. Feasibility testing runs in $O(mu^3)$ time, while full solution computation takes $O(mu^7)$. This improves upon the previous best integral algorithm—whose complexity was $ ilde{O}(mu^{19})$—by an exponential factor, and guarantees integer-valued solutions.

Technology Category

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

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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In this paper we study flow problems on temporal networks, where edge capacities and travel times change over time. We consider a network with $n$ nodes and $m$ edges where the capacity and length of each edge is a piecewise constant function, and use $mu=Omega(m)$ to denote the total number of pieces in all of the $2m$ functions. Our goal is to design exact algorithms for various flow problems that run in time polynomial in the parameter $mu$. Importantly, the algorithms we design are strongly polynomial, i.e. have no dependence on the capacities, flow value, or the time horizon of the flow process, all of which can be exponentially large relative to the other parameters; and return an integral flow when all input parameters are integral. Our main result is an algorithm for checking feasibility of a dynamic transshipment problem on temporal networks -- given multiple sources and sinks with supply and demand values, is it possible to satisfy the desired supplies and demands within a given time horizon? We develop a fast ($O(mu^3)$ time) algorithm for this feasibility problem when the input network has a certain canonical form, by exploiting the cut structure of the associated time expanded network. We then adapt an approach of cite{hoppe2000} to show how other flow problems on temporal networks can be reduced to the canonical format. For computing dynamic transshipments on temporal networks, this results in a $O(mu^7)$ time algorithm, whereas the previous best integral exact algorithm runs in time $ ilde O(mu^{19})$. We achieve similar improvements for other flow problems on temporal networks.

Problem

• Develop exact algorithms for temporal network flows.
• Check feasibility of dynamic transshipment problems.
• Improve time complexity for flow computations.

Innovation

• Strongly polynomial exact algorithms
• Exploiting cut structure in networks
• Canonical form reduction technique