From Incremental Transitive Cover to Strongly Polynomial Maximum Flow
Daniel Dadush, James B. Orlin, Aaron Sidford, László A. Végh
TL;DR
The paper addresses efficient strongly polynomial maximum flow on structured networks by combining a new incremental transitive cover framework with fast approximate flow solvers. At its core, it reduces exact maximum flow to a sequence of approximate solves on compact auxiliary graphs, while maintaining a sparse, updately consistent transitive cover (via witness lists) to amortize rerouting costs. The authors introduce several data-structure variants (ITCO, tree-depth ITCO, and node-ordering ITCO) and prove strong polynomial running times in terms of the number of capacitated arcs $m_c$, the number of nodes $n$, and the matrix-multiplication exponent $ω$, with corollaries for node-capacitated flows, maximum bipartite $b$-matching, and maximum weight closure. For graphs with sublinear treewidth or bounded tree-depth, they obtain substantial speedups, including $O(m^{1+o(1)} D)$ time for tree-depth $D$ and minor-excluded graphs, as well as deterministic $n^{ω+o(1)}$-time solvers in favorable regimes. The framework leverages nearly-linear approximate solvers (via recent FMM-based and randomized methods) and demonstrates how structured extensions and witness-based rerouting can overcome bottlenecks in Orlin’s classical approach, providing a pathway toward faster strongly polynomial flow in important structured families with practical implications for related problems like maximum closure and bipartite matching.
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.