Strongly Polynomial Parallel Maximum Flow Revisited

2026-08-12

AI Summary

This work addresses the efficient parallel computation of maximum flows in directed networks with real-valued capacities. Building upon the strongly polynomial algorithm of Dadush et al., we design a randomized parallel variant that preserves strong polynomiality while significantly improving parallel performance. Our algorithm achieves, for the first time, a balanced trade-off between total work and depth—specifically Õ(mn) work and Õ(m) depth—in the setting of strongly polynomial parallel maximum flow algorithms, thereby overcoming longstanding bottlenecks in either work or depth that plagued prior approaches. Key technical contributions include a novel randomized parallelization strategy, advances in strongly polynomial network flow theory, and efficient parallel graph processing techniques.

Abstract

We study the maximum flow problem in directed networks with real capacities in the parallel setting. For a network with $n$ vertices and $m$ arcs, we show that a randomized parallel implementation of a variant of the strongly polynomial max-flow algorithm of Dadush, Orlin, Sidford, and Végh [SODA 2026] runs in $\tilde{O}(mn)$ work and $\tilde{O}(m)$ depth. This improves upon the previously described tradeoffs between work and depth for strongly polynomial parallel maximum flow algorithms: earlier $\tilde{O}(n^3)$-work algorithms have $\tilde{O}(n^2)$ depth [Shiloach and Vishkin, J. Algorithms 1982; Goldberg and Tarjan, J. ACM 1988], while the known $\tilde{O}(m)$-depth approach uses $\tilde{O}(mn^3)$ work [Orlin, Oper. Res. 1993].

Problem

• maximum flow
• parallel algorithms
• strongly polynomial
• network flow
• computational complexity

Innovation

• maximum flow
• strongly polynomial
• parallel algorithm
• randomized implementation
• work-depth tradeoff