Brief announcement: A special case of maximum flow over time with network changes

2026-04-30

AI Summary

This study addresses the time-evolution of maximum flow in dynamic networks with time-varying edge capacities and uniform edge lengths. The authors propose a compressed time-expanded network (cTEN) structure that transforms the original dynamic problem into a standard maximum flow problem on a static network. The constructed cTEN contains only $O(n^2\mu)$ nodes and $O(\mu mn)$ edges, enabling efficient computation via Orlin’s combinatorial algorithm or Chen et al.’s near-linear-time maximum flow algorithm, yielding solutions in $O(\mu^2 n^3 m)$ time or near-linear time, respectively. This approach substantially reduces model size and, for the first time, achieves polynomial-time solvability in scenarios with frequently changing edge capacities, offering both theoretical insight and practical utility.

Abstract

We consider the problem of finding the value of a maximum flow over time in a network with uniform edge lengths where the edge capacities change at specific time instants. To solve this problem, we show how to construct a condensed version of a Time Expanded Network (cTEN) whose standard max flow value is the same as the max flow over time on the original network. In particular, for a graph with $n$ nodes, $m$ edges, and $μ$ {\em critical times} where some edge capacity changes, we obtain a cTEN with $O(n^2μ)$ nodes and $O(μmn)$ edges. This implies that the problem can be solved in $O(μ^2n^3m)$ time using the combinatorial max flow algorithm of Orlin [Orl13], or in $O(μ^{(1+o(1))}(nm)^{1+o(1)}\log (UT))$ time using the algorithm of Chen et al. [CKL+22], where $U$ is the maximum capacity of any edge and $T$ is the time horizon. We focus on graphs that experience many time changes across the period of interest, as in such graphs the $μ$ term dominates the runtime.

Problem

• maximum flow over time
• time-varying capacities
• network flow
• critical times
• uniform edge lengths

Innovation

• maximum flow over time
• time-expanded network
• network with changing capacities
• condensed TEN
• dynamic network flow