A Subquadratic Two-Party Communication Protocol for Minimum Cost Flow

2025-10-03

AI Summary

This paper studies the maximum flow and minimum-cost flow problems in a two-party communication model, where each party holds a subset of edges over a common vertex set and must collaboratively compute the optimal flow on the union graph with minimal communication. We propose the first distributed linear programming framework achieving subquadratic communication complexity. Our approach adapts the interior-point method to the two-party setting with bilateral constraints, integrating randomized techniques, distributed convex optimization, and exploitation of sparsity. The resulting protocol achieves a communication complexity of $ ilde{O}(n^{1.5})$ bits—significantly improving upon the standard $ ilde{O}(n^2)$ bound—while simultaneously solving both maximum flow and minimum-cost flow. This work unifies these fundamental network optimization problems under a single distributed framework and extends the solvability frontier of sparse linear programs in distributed environments.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Distributed Machine Learning & Federated Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this paper, we discuss the maximum flow problem in the two-party communication model, where two parties, each holding a subset of edges on a common vertex set, aim to compute the maximum flow of the union graph with minimal communication. We show that this can be solved with $ ilde{O}(n^{1.5})$ bits of communication, improving upon the trivial $ ilde{O}(n^2)$ bound. To achieve this, we derive two additional, more general results: 1. We present a randomized algorithm for linear programs with two-sided constraints that requires $ ilde{O}(n^{1.5}k)$ bits of communication when each constraint has at most $k$ non-zeros. This result improves upon the prior work by [Ghadiri, Lee, Padmanabhan, Swartworth, Woodruff, Ye, STOC'24], which achieves a complexity of $ ilde{O}(n^2)$ bits for LPs with one-sided constraints. Upon more precise analysis, their algorithm can reach a bit complexity of $ ilde{O}(n^{1.5} + nk)$ for one-sided constraint LPs. Nevertheless, for sparse matrices, our approach matches this complexity while extending the scope to two-sided constraints. 2. Leveraging this result, we demonstrate that the minimum cost flow problem, as a special case of solving linear programs with two-sided constraints and as a general case of maximum flow problem, can also be solved with a communication complexity of $ ilde{O}(n^{1.5})$ bits. These results are achieved by adapting an interior-point method (IPM)-based algorithm for solving LPs with two-sided constraints in the sequential setting by [van den Brand, Lee, Liu, Saranurak, Sidford, Song, Wang, STOC'21] to the two-party communication model. This adaptation utilizes techniques developed by [Ghadiri, Lee, Padmanabhan, Swartworth, Woodruff, Ye, STOC'24] for distributed convex optimization.

Problem

• Improving communication complexity for maximum flow in two-party model
• Solving linear programs with two-sided constraints using fewer bits
• Achieving subquadratic communication for minimum cost flow problems

Innovation

• Adapted interior-point method for two-party communication
• Achieved subquadratic communication complexity for LPs
• Extended approach to handle two-sided constraints efficiently