Solving Packing and Covering LPs in $\tilde{O}(\frac{1}{\epsilon^2})$ Distributed Iterations with a Single Algorithm and Simpler Analysis

TL;DR

This paper introduces a simple, unified algorithm for solving packing and covering linear programs in distributed settings with an iteration complexity of O(1/psilon^2), providing a more intuitive analysis and highlighting the potential for acceleration.

Contribution

The paper presents a new, straightforward algorithm for packing and covering LPs with optimal iteration complexity, along with an intuitive primal-dual analysis that unifies and simplifies previous approaches.

Findings

Algorithm matches the best known O(1/psilon^2) iteration complexity.

Analysis offers a primal-dual perspective and an approximate optimality gap.

Existing algorithms are unaccelerated, suggesting room for improved accelerated methods.

Abstract

Packing and covering linear programs belong to the narrow class of linear programs that are efficiently solvable in parallel and distributed models of computation, yet are a powerful modeling tool for a wide range of fundamental problems in theoretical computer science, operations research, and many other areas. Following recent progress in obtaining faster distributed and parallel algorithms for packing and covering linear programs, we present a simple algorithm whose iteration count matches the best known $\tilde{O}(\frac{1}{\epsilon^2})$ for this class of problems. The algorithm is similar to the algorithm of [Allen-Zhu and Orecchia, 2015], it can be interpreted as Nesterov's dual averaging, and it constructs approximate solutions to both primal (packing) and dual (covering) problems. However, the analysis relies on the construction of an approximate optimality gap and a primal-dual view, leading to a more intuitive interpretation. Moreover, our analysis suggests that all existing algorithms for solving packing and covering linear programs in parallel/distributed models of computation are, in fact, unaccelerated, and raises the question of designing accelerated algorithms for this class of problems.