Incremental Approximate Maximum Flow via Residual Graph Sparsification

TL;DR

This paper introduces a new incremental algorithm for approximate maximum flow in dense graphs, achieving polylogarithmic update times by extending residual graph sparsification techniques.

Contribution

It extends residual graph sparsification to approximate flows in incremental settings and generalizes cut sparsification to balanced directed graphs.

Findings

Achieves polylogarithmic amortized update time for dense graphs.

Maintains a $(1-
-)$-approximate $s$-$t$ maximum flow efficiently.

First to handle such updates in dense graphs with high probability.

Abstract

We give an algorithm that, with high probability, maintains a $(1-\epsilon)$-approximate $s$-$t$ maximum flow in undirected, uncapacitated $n$-vertex graphs undergoing $m$ edge insertions in $\tilde{O}(m+ n F^*/\epsilon)$ total update time, where $F^{*}$ is the maximum flow on the final graph. This is the first algorithm to achieve polylogarithmic amortized update time for dense graphs ($m = \Omega(n^2)$), and more generally, for graphs where $F^*= \tilde{O}(m/n)$. At the heart of our incremental algorithm is the residual graph sparsification technique of Karger and Levine [STOC '02, SICOMP '15], originally designed for computing exact maximum flows in the static setting. Our main contributions are (i) showing how to maintain such sparsifiers for approximate maximum flows in the incremental setting and (ii) generalizing the cut sparsification framework of Fung et al. [STOC '11, SICOMP '19] from undirected graphs to balanced directed graphs.