Generalized Flow in Nearly-linear Time on Moderately Dense Graphs

TL;DR

This paper introduces a nearly-linear time randomized algorithm for solving generalized flow problems in dense graphs, significantly improving efficiency over previous methods by leveraging new data structures and spectral analysis within an interior point framework.

Contribution

It presents the first nearly-linear time algorithm for generalized flow problems on dense graphs, using novel dynamic data structures and spectral techniques.

Findings

Achieved a nearly-linear time complexity of tilde;(m + n^{1.5}) polylog(W/) for generalized flows.

Improved upon previous algorithms with tilde;(m  tilde;(m  tilde;(m + n^{1.5}) polylog(W/)).

Developed new spectral results and dynamic data structures for generalized flow matrices.

Abstract

In this paper we consider generalized flow problems where there is an $m$-edge $n$-node directed graph $G = (V,E)$ and each edge $e \in E$ has a loss factor $\gamma_e >0$ governing whether the flow is increased or decreased as it crosses edge $e$. We provide a randomized $\tilde{O}( (m + n^{1.5}) \cdot \mathrm{polylog}(\frac{W}{\delta}))$ time algorithm for solving the generalized maximum flow and generalized minimum cost flow problems in this setting where $\delta$ is the target accuracy and $W$ is the maximum of all costs, capacities, and loss factors and their inverses. This improves upon the previous state-of-the-art $\tilde{O}(m \sqrt{n} \cdot \log^2(\frac{W}{\delta}) )$ time algorithm, obtained by combining the algorithm of [Daitch-Spielman, 2008] with techniques from [Lee-Sidford, 2014]. To obtain this result we provide new dynamic data structures and spectral results regarding the matrices associated to generalized flows and apply them through the interior point method framework of [Brand-Lee-Liu-Saranurak-Sidford-Song-Wang, 2021].