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Combinatorial Maximum Flow via Weighted Push-Relabel on Shortcut Graphs

Aaron Bernstein, Joakim Blikstad, Jason Li, Thatchaphol Saranurak, Ta-Wei Tu

TL;DR

This work delivers a purely combinatorial maximum-flow algorithm for directed graphs with capacities bounded by $U$, achieving a near-quadratic time bound $\tilde{O}(n^{2}\log^{19}n\log U)$ and reducing the gap to prior dense-graph results. The core idea is to run a weighted push-relabel procedure on a shortcut graph that augments the original graph with level-specific star shortcuts derived from a weak expander hierarchy, enabling independent, low-cost routing across levels. A key contribution is showing that shortcuts restore the short-flow property even for weak hierarchies, simplifying both the construction of the hierarchy and the analysis, and enabling an efficient bottom-up expansion strategy. The algorithm is randomized, but the authors give a deterministic near-linear-time variant for vertex-capacitated max flow, implying the first such near-linear deterministic algorithm in dense regimes; a full implementation further demonstrates the practical clarity of the approach. Overall, the combination of weighted push-relabel, shortcut-augmented expanders, and a streamlined expander-decomposition framework yields a simpler, provably faster, and implementable path beyond the classical $\tilde{O}(m\sqrt{n})$ barrier for general graphs.

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 $\tilde{O}(n^{2}\log U)$ 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 $\tilde{O}(m\sqrt{n})$ 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 $\tilde{O}(n^2)$ 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.

Combinatorial Maximum Flow via Weighted Push-Relabel on Shortcut Graphs

TL;DR

This work delivers a purely combinatorial maximum-flow algorithm for directed graphs with capacities bounded by , achieving a near-quadratic time bound and reducing the gap to prior dense-graph results. The core idea is to run a weighted push-relabel procedure on a shortcut graph that augments the original graph with level-specific star shortcuts derived from a weak expander hierarchy, enabling independent, low-cost routing across levels. A key contribution is showing that shortcuts restore the short-flow property even for weak hierarchies, simplifying both the construction of the hierarchy and the analysis, and enabling an efficient bottom-up expansion strategy. The algorithm is randomized, but the authors give a deterministic near-linear-time variant for vertex-capacitated max flow, implying the first such near-linear deterministic algorithm in dense regimes; a full implementation further demonstrates the practical clarity of the approach. Overall, the combination of weighted push-relabel, shortcut-augmented expanders, and a streamlined expander-decomposition framework yields a simpler, provably faster, and implementable path beyond the classical barrier for general graphs.

Abstract

We give a combinatorial algorithm for computing exact maximum flows in directed graphs with vertices and edge capacities from in time, which is near-optimal on dense graphs. This shaves an 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 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 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.
Paper Structure (60 sections, 35 theorems, 26 equations, 3 algorithms)

This paper contains 60 sections, 35 theorems, 26 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Augmenting Paths: The Impact of Clarity.
  3. Interior Point Methods and Dynamic Graph Data Structures.
  4. In Pursuit of Clarity.
  5. Our Result.
  6. Implementation.
  7. Derandomization.
  8. Techniques.
  9. Preliminaries
  10. Basic Definitions
  11. Graphs.
  12. Incident Edges, Degree, Volume, and Cuts.
  13. Flows.
  14. Weighted Push-Relabel
  15. Crucial Concepts: Hierarchies, Expanders, and Shortcuts
  16. ...and 45 more sections

Key Result

Theorem 1.1

There is a randomized algorithm that, given an $n$-vertex directed graph $G$ with edge capacities from $\{1,\dots,U\}$ and $s,t\in V(G)$, finds a maximum $(s,t)$-flow in $G$ with high probability in $O(n^{2}\log^{19}n\log U)$ time.

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • lemma 2.1: Flow rounding KangP15
  • theorem 2.2: Weighted Push-Relabel BernsteinBST24
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: Weak Expander Hierarchy
  • Definition 2.7
  • Definition 2.8: Shortcut of Hierarchy
  • ...and 68 more