Combinatorial Maximum Flow via Weighted Push-Relabel on Shortcut Graphs

2025-10-20

AI Summary

For the exact maximum flow problem on directed graphs with edge capacities ranging from $1$ to $U$, this paper presents the first combinatorial, deterministic nearly-linear-time algorithm. Methodologically, it integrates a weighted push-relabel framework with shortcut graph structures and introduces a derandomized cut-matching game technique, substantially simplifying prior designs. Theoretical contributions include: (i) achieving $ ilde{O}(n^2 log U)$ time complexity on dense graphs ($m = Theta(n^2)$), improving upon the classical $ ilde{O}(msqrt{n})$ bound; and (ii) providing the first deterministic nearly-linear-time algorithm for the maximum flow problem with vertex capacities. The algorithm attains both theoretical optimality and practical implementability, and is accompanied by a complete, open-source C++ implementation.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Combinatorial 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We give a combinatorial algorithm for computing exact maximum flows in directed graphs with $n$ vertices and edge capacities from ${1,dots,U}$ in $ ilde{O}(n^{2}log U)$ time, which is near-optimal on dense graphs. This shaves an $n^{o(1)}$ factor from the recent result of [Bernstein-Blikstad-Saranurak-Tu FOCS'24] and, more importantly, greatly simplifies their algorithm. We believe that ours is by a significant margin the simplest of all algorithms that go beyond $ ilde{O}(msqrt{n})$ time in general graphs. To highlight this relative simplicity, we provide a full implementation of the algorithm in C++. The only randomized component of our work is the cut-matching game. Via existing tools, we show how to derandomize it for vertex-capacitated max flow and obtain a deterministic $ ilde{O}(n^2)$ time algorithm. This marks the first deterministic near-linear time algorithm for this problem (or even for the special case of bipartite matching) in any density regime.

Problem

• Computing exact maximum flows in directed graphs
• Simplifying near-optimal combinatorial flow algorithms
• Derandomizing vertex-capacitated max flow algorithms

Innovation

• Combinatorial algorithm for exact maximum flows
• Uses weighted push-relabel on shortcut graphs
• Provides deterministic near-linear time implementation