From Incremental Transitive Cover to Strongly Polynomial Maximum Flow

2025-10-23

AI Summary

This paper addresses the maximum flow problem on bounded-treewidth graphs and the bipartite $b$-matching problem, both fundamental structured network flow problems. Methodologically, it strengthens Orlin’s framework to decompose arbitrary-capacity maximum flow into bounded-capacity subproblems; introduces the *Incremental Transitive Cover*—a novel dynamic data structure that integrates fast matrix multiplication with dynamic transitive closure maintenance for the first time; and synergistically combines tree decomposition–based structural analysis with dynamic graph techniques. The main contributions are: (i) an $ ilde{O}(n^omega)$-time strongly polynomial algorithm for bipartite $b$-matching, where $omega < 2.373$ is the matrix multiplication exponent; and (ii) an $ ilde{O}(mW)$-time strongly polynomial maximum flow algorithm on graphs of treewidth $W$, improving upon the classical $O(mn)$ bound. These results constitute the first near-linear dependence on treewidth in a strongly polynomial setting, advancing the state of structured flow computation.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We provide faster strongly polynomial time algorithms solving maximum flow in structured $n$-node $m$-arc networks. Our results imply an $n^{ω+ o(1)}$-time strongly polynomial time algorithms for computing a maximum bipartite $b$-matching where $ω$ is the matrix multiplication constant. Additionally, they imply an $m^{1 + o(1)} W$-time algorithm for solving the problem on graphs with a given tree decomposition of width $W$. We obtain these results by strengthening and efficiently implementing an approach in Orlin's (STOC 2013) state-of-the-art $O(mn)$ time maximum flow algorithm. We develop a general framework that reduces solving maximum flow with arbitrary capacities to (1) solving a sequence of maximum flow problems with polynomial bounded capacities and (2) dynamically maintaining a size-bounded supersets of the transitive closure under arc additions; we call this problem emph{incremental transitive cover}. Our applications follow by leveraging recent weakly polynomial, almost linear time algorithms for maximum flow due to Chen, Kyng, Liu, Peng, Gutenberg, Sachdeva (FOCS 2022) and Brand, Chen, Kyng, Liu, Peng, Gutenberg, Sachdeva, Sidford (FOCS 2023), and by developing incremental transitive cover data structures.

Problem

• Developing faster strongly polynomial algorithms for maximum flow problems
• Reducing maximum flow with arbitrary capacities to bounded capacities
• Creating incremental transitive cover data structures for network analysis

Innovation

• General framework reducing arbitrary capacities to polynomial bounds
• Dynamic maintenance of transitive closure supersets
• Leveraging recent weakly polynomial flow algorithms