Parallel $(1+\epsilon)$-Approximate Multi-Commodity Mincost Flow in Almost Optimal Depth and Work

TL;DR

This paper introduces a nearly optimal parallel algorithm for approximate multi-commodity mincost flow that achieves low work and depth, generalizing previous special-case algorithms and introducing new flow shortcut constructions.

Contribution

It presents the first construction of length-constrained flow shortcuts with controlled slack and congestion, enabling almost optimal parallel algorithms for multi-commodity mincost flow.

Findings

Achieves (m) work and (1) depth for approximate mincost flow.

Generalizes previous algorithms to handle vertex capacities and multiple commodities.

Introduces new flow shortcut techniques with controlled length and congestion slack.

Abstract

We present a parallel algorithm for computing $(1+\epsilon)$-approximate mincost flow on an undirected graph with $m$ edges, where capacities and costs are assigned to both edges and vertices. Our algorithm achieves $\hat{O}(m)$ work and $\hat{O}(1)$ depth when $\epsilon > 1/\mathrm{polylog}(m)$, making both the work and depth almost optimal, up to a subpolynomial factor. Previous algorithms with $\hat{O}(m)$ work required $\Omega(m)$ depth, even for special cases of mincost flow with only edge capacities or max flow with vertex capacities. Our result generalizes prior almost-optimal parallel $(1+\epsilon)$-approximation algorithms for these special cases, including shortest paths [Li, STOC'20] [Andoni, Stein, Zhong, STOC'20] [Rozhen, Haeupler, Marinsson, Grunau, Zuzic, STOC'23] and max flow with only edge capacities [Agarwal, Khanna, Li, Patil, Wang, White, Zhong, SODA'24]. Our key technical contribution is the first construction of length-constrained flow shortcuts with $(1+\epsilon)$ length slack, $\hat{O}(1)$ congestion slack, and $\hat{O}(1)$ step bound. This provides a strict generalization of the influential concept of $(\hat{O}(1),\epsilon)$-hopsets [Cohen, JACM'00], allowing for additional control over congestion. Previous length-constrained flow shortcuts [Haeupler, Hershkowitz, Li, Roeyskoe, Saranurak, STOC'24] incur a large constant in the length slack, which would lead to a large approximation factor. To enable our flow algorithms to work under vertex capacities, we also develop a close-to-linear time algorithm for computing length-constrained vertex expander decomposition. Building on Cohen's idea of path-count flows [Cohen, SICOMP'95], we further extend our algorithm to solve $(1+\epsilon)$-approximate $k$-commodity mincost flow problems with almost-optimal $\hat{O}(mk)$ work and $\hat{O}(1)$ depth, independent of the number of commodities $k$.