An Efficient Mean Field Approach to the Set Covering Problem

TL;DR

This paper introduces a mean field neural network algorithm with a multilinear penalty function for efficiently solving large set covering problems, achieving near-optimal solutions with minimal computational resources.

Contribution

It develops a novel mean field feedback neural network approach with a multilinear penalty for the set covering problem, enabling fast approximate solutions for large instances.

Findings

Achieves solutions within a few percent of optimal for test problems.

Requires only a few seconds CPU time per problem.

Successfully scales to large problem sizes up to 5,000 rows and 1,000,000 columns.

Abstract

A mean field feedback artificial neural network algorithm is developed and explored for the set covering problem. A convenient encoding of the inequality constraints is achieved by means of a multilinear penalty function. An approximate energy minimum is obtained by iterating a set of mean field equations, in combination with annealing. The approach is numerically tested against a set of publicly available test problems with sizes ranging up to 5x10^3 rows and 10^6 columns. When comparing the performance with exact results for sizes where these are available, the approach yields results within a few percent from the optimal solutions. Comparisons with other approximate methods also come out well, in particular given the very low CPU consumption required -- typically a few seconds. Arbitrary problems can be processed using the algorithm via a public domain server.