Maximum Flow by Augmenting Paths in $n^{2+o(1)}$ Time

2024-06-05

IEEE Annual Symposium on Foundations of Computer Science

AI Summary

This paper studies the maximum flow problem on directed graphs with integer capacities (1 leq c(e) leq U), aiming to break the long-standing time lower bound for combinatorial algorithms on dense graphs. We propose the first purely combinatorial algorithm that integrates a hierarchical directed expander decomposition with a weighted push-relabel framework, achieving an exact running time of (O(n^{2+o(1)} log U)) on graphs with (n) vertices and (m) edges. The algorithm avoids continuous optimization and dynamic data structures, significantly accelerating augmenting path enumeration. For unit-capacity dense graphs with (m = omega(n^{4/3})), it improves upon Karzanov’s and Even–Tarjan’s algorithms, running in (n^{2+o(1)}) time—matching the best-known combinatorial bounds for optimal bipartite matching and all prior combinatorial results except Chen et al. (FOCS’22). This is the first combinatorial algorithm to approach the theoretical optimum for dense-graph maximum flow.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We present a combinatorial algorithm for computing exact maximum flows in directed graphs with $n$ vertices and edge capacities from ${1, ldots, U}$ in $n^{2+o(1)}log U$ time, which is almost optimal in dense graphs. Our algorithm is a novel implementation of the classical augmenting-path framework; we list augmenting paths more efficiently using a new variant of the push-relabel algorithm that uses additional edge weights to guide the algorithm, and we derive the edge weights by constructing a directed expander hierarchy. Even in unit-capacity graphs, this breaks the long-standing $O(m cdotmin{sqrt{m},n^{2/3}})$ time bound of the previous combinatorial algorithms by Karzanov (1973) and Even and Tarjan (1975) when the graph has $m=omega(n^{4/3})$ edges. Notably, our approach does not rely on continuous optimization nor heavy dynamic graph data structures, both of which are crucial in the recent developments that led to the almost-linear time algorithm by Chen et al. (FOCS 2022). Our running time also matches the $n^{2+o(1)}$ time bound of the independent combinatorial algorithm by Chuzhoy and Khanna (STOC 2024) for computing the maximum bipartite matching, a special case of maximum flow.

Problem

• Computing exact maximum flows in directed graphs efficiently
• Breaking long-standing time bounds for combinatorial flow algorithms
• Achieving near-optimal runtime without continuous optimization methods

Innovation

• Uses push-relabel variant with edge weights
• Guides algorithm via directed expander hierarchy
• Achieves near-optimal time without continuous optimization