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

TL;DR

This paper introduces a method to efficiently compute maximum flow over time in networks with capacity changes at specific times by constructing a condensed network, enabling faster algorithms.

Contribution

It presents a novel construction of a condensed Time Expanded Network (cTEN) that preserves max flow values, improving computational efficiency for networks with many capacity change points.

Findings

Constructed cTEN with O(n^2μ) nodes and O(μmn) edges.

Achieved a max flow computation time of O(μ^2n^3m) using Orlin's algorithm.

Provided an alternative faster algorithm with time complexity O(μ^{(1+o(1))}(nm)^{1+o(1)} log(UT)).

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 $\mu$ {\em critical times} where some edge capacity changes, we obtain a cTEN with $O(n^2\mu)$ nodes and $O(\mu mn)$ edges. This implies that the problem can be solved in $O(\mu^2n^3m)$ time using the combinatorial max flow algorithm of Orlin [Orl13], or in $O(\mu^{(1+o(1))}(nm)^{1+o(1)}\log (UT))$ 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 $\mu$ term dominates the runtime.