Efficient parallel implementation of the multiplicative weight update method for graph-based linear programs

TL;DR

This paper presents an efficient parallel implementation of the multiplicative weight update method for solving positive linear programs related to graph problems, demonstrating superior speed and scalability compared to existing solvers and algorithms.

Contribution

The authors develop a novel parallel algorithm with heuristic step size search and sparse linear algebra optimizations, improving practical performance for graph-based LP relaxations.

Findings

Faster than general LP solvers like CPLEX and Gurobi

Outperforms specialized parallel graph algorithms in some cases

Demonstrates good scalability on supercomputing hardware

Abstract

Positive linear programs (LPs) model many graph and operations research problems. One can solve for a $(1+\epsilon)$-approximation for positive LPs, for any selected $\epsilon$, in polylogarithmic depth and near-linear work via variations of the multiplicative weight update (MWU) method. Despite extensive theoretical work on these algorithms through the decades, their empirical performance is not well understood. In this work, we implement and test an efficient parallel algorithm for solving positive LP relaxations, and apply it to graph problems such as densest subgraph, bipartite matching, vertex cover and dominating set. We accelerate the algorithm via a new step size search heuristic. Our implementation uses sparse linear algebra optimization techniques such as fusion of vector operations and use of sparse format. Furthermore, we devise an implicit representation for graph incidence constraints. We demonstrate the parallel scalability with the use of threading OpenMP and MPI on the Stampede2 supercomputer. We compare this implementation with exact libraries and specialized libraries for the above problems in order to evaluate MWU's practical standing for both accuracy and performance among other methods. Our results show this implementation is faster than general purpose LP solvers (IBM CPLEX, Gurobi) in all of our experiments, and in some instances, outperforms state-of-the-art specialized parallel graph algorithms.