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A Subquadratic Two-Party Communication Protocol for Minimum Cost Flow

Hossein Gholizadeh, Yonggang Jiang

TL;DR

This work analyzes the two-party communication complexity of classic flow problems, showing that maximum flow and minimum cost flow can be solved with subquadratic communication in the two-party model. The authors adapt a sequential interior-point method for linear programs with two-sided constraints to the distributed setting, combining $oldsymbol{ extell}_p$-Lewis weights, spectral approximation, and inverse-maintenance techniques to bound communication. They prove a main LP result with $ ilde{O}ig(n^{1.5}L^2(k+ ext{log} abla ext{log} m) ext{log} mig)$ bits and derive a mincost-flow protocol with $ ilde{O}ig(n^{1.5} ext{log}^2( orm{u}_ inf orm{c}_ inf)ig)$ bits, with maxflow obtained via a standard reduction and subsequent scaling. The results close a gap between trivial transmission bounds and subquadratic benchmarks, offering practical implications for distributed network optimization and related computational models.

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 $\tilde{O}(n^{1.5})$ bits of communication, improving upon the trivial $\tilde{O}(n^2)$ 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 $\tilde{O}(n^{1.5}k)$ 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 $\tilde{O}(n^2)$ bits for LPs with one-sided constraints. Upon more precise analysis, their algorithm can reach a bit complexity of $\tilde{O}(n^{1.5} + nk)$ 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 $\tilde{O}(n^{1.5})$ 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.

A Subquadratic Two-Party Communication Protocol for Minimum Cost Flow

TL;DR

This work analyzes the two-party communication complexity of classic flow problems, showing that maximum flow and minimum cost flow can be solved with subquadratic communication in the two-party model. The authors adapt a sequential interior-point method for linear programs with two-sided constraints to the distributed setting, combining -Lewis weights, spectral approximation, and inverse-maintenance techniques to bound communication. They prove a main LP result with bits and derive a mincost-flow protocol with bits, with maxflow obtained via a standard reduction and subsequent scaling. The results close a gap between trivial transmission bounds and subquadratic benchmarks, offering practical implications for distributed network optimization and related computational models.

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 bits of communication, improving upon the trivial 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 bits of communication when each constraint has at most non-zeros. This result improves upon the prior work by [Ghadiri, Lee, Padmanabhan, Swartworth, Woodruff, Ye, STOC'24], which achieves a complexity of bits for LPs with one-sided constraints. Upon more precise analysis, their algorithm can reach a bit complexity of 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 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.

Paper Structure

This paper contains 32 sections, 16 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. A concurrent work BrandWZ25.
  4. Organization
  5. Preliminaries
  6. Matrices.
  7. Matrix and Vector Operations.
  8. Highly 1-Self-Concordant Barrier Functions $\phi_i$.
  9. Leverage Scores and Spectral Approximation.
  10. $\ell_p$-Lewis Weights.
  11. Bit Complexity.
  12. Two-Party Communication Model.
  13. Technical Overview
  14. Linear Programming in the Sequential Setting
  15. Intuition Behind the Approach.
  16. ...and 17 more sections

Key Result

Theorem 1.1

For any constant $\delta > 0$, there exists a randomized algorithm in the two-party communication setting that, given $\mathbf{A} \in \mathbb{R}^{m \times n}$, $c, \ell, u \in \mathbb{R}^m$, and $b \in \mathbb{R}^n$, and assuming there exists a point $x \in \mathbb{R}^m$ such that $\mathbf{A}^\top x The algorithm requires where $k = \max_{i \in [m]}(\operatorname{nnz}(a_i))$ is the upper bound on

Theorems & Definitions (29)

  • Theorem 1.1: Linear Programming in the Two-Party Communication Model
  • Theorem 1.2: Minimum Cost Flow in the Two-Party Communication Model
  • Theorem 1.3: Maximum Flow in the Two-Party Communication Model
  • Theorem 4.1: Linear Programming in the Two-Party Communication Model
  • Lemma 4.1: Adapted from lemma 4.7 of ghadiri2024improvingbitcomplexitycommunication
  • Lemma 4.2: Approximating Leverage Scores, Adapted from lemma 3.2 of ghadiri2024improvingbitcomplexitycommunication
  • Definition 4.3: Sampling Function, Definition 3.3 of ghadiri2024improvingbitcomplexitycommunication
  • Lemma 4.4: Spectral Approximation via Leverage Score, partially adopted from lemma 3.4 of ghadiri2024improvingbitcomplexitycommunication
  • proof : Proof of \ref{['lemma:specApproxlemma']}
  • Lemma 4.5: Inverse Maintenance, Algorithm 10 of ghadiri2024improvingbitcomplexitycommunication
  • ...and 19 more