Generalized Flow in Nearly-linear Time on Moderately Dense Graphs

2025-10-20

AI Summary

This paper studies the generalized flow problem—including generalized maximum flow and minimum-cost flow—on directed graphs, where each edge is associated with a gain/loss factor. For moderately dense graphs, we present the first nearly-linear-time algorithm. Our approach builds upon an interior-point method framework, integrating randomized techniques, dynamic graph data structures, and spectral graph theory to achieve a key breakthrough in the analysis of generalized flow-related matrices. The algorithm runs in $ ilde{O}((m + n^{1.5}) cdot mathrm{polylog}(W/delta))$ time, improving upon the previous best $ ilde{O}(msqrt{n})$ bound and yielding the first substantial speedup for moderately dense graphs. Here, $m$ and $n$ denote the number of edges and vertices, respectively, while $W$ and $delta$ represent standard problem-dependent parameters governing capacity and accuracy requirements.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $γ_e >0$ governing whether the flow is increased or decreased as it crosses edge $e$. We provide a randomized $ ilde{O}( (m + n^{1.5}) cdot mathrm{polylog}(frac{W}δ))$ time algorithm for solving the generalized maximum flow and generalized minimum cost flow problems in this setting where $δ$ 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 $ ilde{O}(m sqrt{n} cdot log^2(frac{W}δ) )$ 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].

Problem

• Solving generalized maximum flow in near-linear time
• Developing randomized algorithm for generalized minimum cost flow
• Improving computational efficiency for moderately dense graphs

Innovation

• Randomized algorithm for generalized flow problems
• Dynamic data structures for interior point method
• Spectral analysis of generalized flow matrices