Gradient descent procedure for solving linear programming relaxations of combinatorial optimization problems in parallel mode on extra large scale

TL;DR

This paper introduces a parallel gradient descent algorithm for solving large-scale LP relaxations of combinatorial optimization problems efficiently on GPUs, significantly outperforming traditional methods like simplex.

Contribution

The paper presents a novel GPU-accelerated gradient descent method tailored for LP relaxations in combinatorial optimization, enabling rapid solutions for large problem instances.

Findings

Solves a 100,000-node fractional 2-matching problem in seconds on GPU.

Outperforms simplex method, which takes over an hour for the same problem.

Algorithm can be adapted for more complex LP relaxations.

Abstract

Linear programming (LP) relaxation is a standard technique for solving hard combinatorial optimization (CO) problems. Here we present a gradient descent algorithm which exploits the special structure of some LP relaxations induced by CO problems. The algorithm can be run in parallel mode and was implemented as CUDA C/C++ program to be executed on GPU. We exemplify efficiency of the algorithm by solving a fractional 2-matching problem. Our results demonstrate that a fractional 2-matching problem with 100,000 nodes is solved by our algorithm on a modern GPU on a scale of a second while solving the problem with simplex method would take more than an hour. The algorithm can be modified to solve more complicated LP relaxations derived from CO problems.