Parallel Minimum Cost Flow in Near-Linear Work and Square Root Depth for Dense Instances

TL;DR

The paper addresses the challenge of solving the exact minimum-cost flow problem in parallel with near-linear work and sublinear depth on dense graphs. It achieves this via a randomized parallel interior-point method augmented with a dynamic parallel expander-decomposition that supports low-depth batch updates, enabling efficient Newton steps. The main result is a parallel algorithm that runs in $ ilde{O}(m+n^{1.5})$ work and $ ilde{O}(\,\sqrt{n}\,)$ depth for $m\ge n^{1.5}$, with corollaries including max flow, bipartite matching, negative-weight shortest paths, and reachability. The key technical advances are dynamic expander decomposition, parallel unit-flow primitives, trimming, and leverage-weighted sampling data structures, which together bring near-linear work and sublinear depth to multiple dense-graph problems. This work advances practical parallel solvers for fundamental network-flow problems and highlights the utility of expander-based decompositions in parallel optimization.

Abstract

For $n$-vertex $m$-edge graphs with integer polynomially-bounded costs and capacities, we provide a randomized parallel algorithm for the minimum cost flow problem with $\tilde O(m+n^ {1.5})$ work and $\tilde O(\sqrt{n})$ depth. On moderately dense graphs ($m>n^{1.5}$), our algorithm is the first one to achieve both near-linear work and sub-linear depth. Previous algorithms are either achieving almost optimal work but are highly sequential [Chen, Kyng, Liu, Peng, Gutenberg, Sachdev, FOCS'22], or achieving sub-linear depth but use super-linear work, [Lee, Sidford, FOCS'14], [Orlin, Stein, Oper. Res. Lett.'93]. Our result also leads to improvements for the special cases of max flow, bipartite maximum matching, shortest paths, and reachability. Notably, the previous algorithms achieving near-linear work for shortest paths and reachability all have depth $n^{o(1)}\cdot \sqrt{n}$ [Fischer, Haeupler, Latypov, Roeyskoe, Sulser, SOSA'25], [Liu, Jambulapati, Sidford, FOCS'19]. Our algorithm consists of a parallel implementation of [van den Brand, Lee, Liu, Saranurak, Sidford, Song, Wang, STOC'21]. One important building block is a \emph{dynamic} parallel expander decomposition, which we show how to obtain from the recent parallel expander decomposition of [Chen, Meierhans, Probst Gutenberh, Saranurak, SODA'25].

Theorems & Definitions (70)

• Theorem 1.1: Informal
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Lemma 1.6: Informal version of \ref{['lem:dynamicExpanderDecomposition']}
• Lemma 3.1
• Theorem 3.2: Parallel Expander Decomposition, Theorem 1 of ChenMGS25
• Lemma 3.3: Parallel expander pruning
• proof : Proof of \ref{['lem:dynamicExpanderDecomposition']}