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

2025-03-17

AI Summary

This paper addresses the minimum-cost flow problem on dense graphs with integer capacities ($m > n^{1.5}$), presenting the first randomized parallel algorithm achieving both near-linear total work $ ilde{O}(m + n^{1.5})$ and sublinear depth $ ilde{O}(sqrt{n})$. Methodologically, it pioneers the integration of van den Brand et al.’s (STOC’21) interior-point method framework with Chen et al.’s (SODA’25) parallel expander decomposition, introducing novel techniques: dynamic parallel expander decomposition, randomized path-following, and dynamic graph partitioning. This synthesis overcomes longstanding parallel efficiency bottlenecks for dense graphs. As a consequence, the algorithm improves the parallel complexity—particularly the depth—of several fundamental problems, including maximum flow, bipartite matching, single-source shortest paths, and reachability. Crucially, it reduces the depth for these tasks from $n^{o(1)}sqrt{n}$ to $ ilde{O}(sqrt{n})$, representing a significant theoretical advance in parallel graph algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Search and Optimization: Combinatorial 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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $ ilde O(m+n^ {1.5})$ work and $ ilde 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].

Problem

• Efficient parallel algorithm for minimum cost flow
• Achieves near-linear work and sub-linear depth
• Improves performance for max flow and shortest paths

Innovation

• Randomized parallel algorithm for minimum cost flow
• Dynamic parallel expander decomposition technique
• Achieves near-linear work and sub-linear depth