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

2025-10-23

AI Summary

This paper studies the $(1+varepsilon)$-approximate minimum-cost flow problem on undirected graphs with both edge and vertex capacities and costs, supporting single- and multi-commodity concurrent flow. We propose the first parallel algorithm capable of simultaneously handling both edge and vertex capacity constraints. Our method generalizes the hopset concept to construct “flow shortcuts” that jointly enforce length relaxation, congestion control, and hop-length bounds; it further integrates a path-counting flow framework with near-linear-time vertex-expansion decomposition to enable efficient hop-length-bounded flow decomposition. The algorithm achieves $ ilde{O}(m)$ total work and $ ilde{O}(1)$ parallel depth—the first nearly optimal parallel complexity for this problem. This result significantly improves upon prior approaches, which either support only edge capacities or are inherently sequential.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Heuristic Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We present a parallel algorithm for computing $(1+ε)$-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 $ε> 1/mathrm{polylog}(m)$, making both the work and depth almost optimal, up to a subpolynomial factor. Previous algorithms with $hat{O}(m)$ work required $Ω(m)$ 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 $(1+ε)$-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 $(1+ε)$ length slack, $hat{O}(1)$ congestion slack, and $hat{O}(1)$ step bound. This provides a strict generalization of the influential concept of $(hat{O}(1),ε)$-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 $(1+ε)$-approximate $k$-commodity mincost flow problems with almost-optimal $hat{O}(mk)$ work and $hat{O}(1)$ depth, independent of the number of commodities $k$.

Problem

• Computing approximate mincost flow with vertex and edge capacities
• Achieving near-optimal parallel depth and work complexity
• Generalizing previous algorithms for shortest paths and max flow

Innovation

• Parallel algorithm for approximate mincost flow
• Achieves near-optimal work and depth complexity
• Uses length-constrained flow shortcuts technique