The Minor inequalities in the description of the Set Covering Polyhedron of Circulant Matrices

TL;DR

This paper provides a complete linear inequality description of the set covering polyhedron for certain circulant matrices, revealing that all non-boolean facets relate to circulant minors and offering an efficient separation algorithm.

Contribution

It offers a full characterization of the set covering polyhedron for specific circulant matrices, linking non-boolean facets to circulant minors and providing a polynomial-time separation method.

Findings

All non-boolean facet inequalities correspond to circulant minors.

A polynomial-time algorithm for separating these inequalities is developed.

The description applies to matrices with parameters s=2,3 and k≥3.

Abstract

In this work we give a complete description of the set covering polyhedron of circulant matrices $C^k_{sk}$ with $s = 2,3$ and $k\geq 3 $ by linear inequalities. In particular, we prove that every non boolean facet defining inequality is associated with a circulant minor of the matrix. We also give a polynomial time separation algorithm for inequalities involved in the description.