A-priori Upper Bounds for the Set Covering Problem

TL;DR

This paper introduces a new probabilistic upper bound for the Set Covering problem with unit costs, applicable to fixed-dimension problems and dependent on matrix size and row densities, also considering block decomposable matrices.

Contribution

It presents a novel probabilistic bound for fixed-dimension Set Covering problems, extending asymptotic results and analyzing matrix decompositions for improved bounds.

Findings

New probabilistic upper bound for fixed-dimension problems

Bound depends on matrix size and row densities

Improved bounds for block decomposable matrices

Abstract

In this paper we present a new bound obtained with the probabilistic method for the solution of the Set Covering problem with unit costs. The bound is valid for problems of fixed dimension, thus extending previous similar asymptotic results, and it depends only on the number of rows of the coefficient matrix and the row densities. We also consider the particular case of matrices that are \textit{almost} block decomposable, and show how the bound may improve according to the particular decomposition adopted. Such final result may provide interesting indications for comparing different matrix decomposition strategies.