# Requiring Connectivity in the Set Covering Problem

## Abstract

Given a bipartite graph with bipartition *V* and *W*, a cover is a subset *C* \({\subseteq}\) *V* such that each node of *W* is adjacent to at least one node in *C*. The set covering problem seeks a minimum cardinality cover. Set covering has many practical applications. In the context of reserve selection for conservation of species, *V* is a set of candidate sites from a reserve network, *W* is the set of species to be protected, and the edges describe which species are represented in each site. Some covers however may assume spatial configurations which are not adequate for conservational purposes. Indeed, for sustainability reasons the fragmentation of existing natural habitats should be avoided, since this is recognized as being disruptive to the species adapted to the habitats. Thus, connectivity appears to be an important issue for protection of biological diversity. We therefore consider along with the bipartite graph, a graph *G* with node set *V*, describing the adjacencies of the elements of *V*, and we look for those covers *C* \({\subseteq}\) *V* for which the subgraph of *G* induced by *C* is connected. We call such covers connected covers. In this paper we introduce and study some valid inequalities for the convex hull of the set of incidence vectors of connected covers.

## Keywords

set covering graphs connected components integer polytopes## Preview

Unable to display preview. Download preview PDF.

## References

- E. Balas and S.M. Ng “On the set covering polytope: I. all facets with coefficients in {0,1,2},”
*Mathematical Programming*, vol. 43, pp. 57–69,1989a.Google Scholar - E. Balas and S.M. Ng “On the set covering polytope: II. lifting the facets with coefficients in {0,1,2},”
*Mathematical Programming*, vol. 45, pp. 1–20, 1989b.Google Scholar - R.A. Briers “Incorporating connectivity into reserve selection procedures,”
*Biological Conservation*, vol. 103, pp. 77–83, 2002.CrossRefGoogle Scholar - D.M. Debinski and R.D. Holt “A survey and overviewof habitat fragmentation,”
*Conservation Biology*, vol. 14, pp. 342–355, 2000.CrossRefGoogle Scholar - G. Cornuéjols and A. Sassano “On the 0,1 facets of the set covering polytope,”
*Mathematical Programming*, vol. 43, pp. 45–55, 1989.Google Scholar - P.L. Hammer, E.L. Johnson, and U.N. Peled “Facets of regular 0-1 polytopes,”
*Mathematical Programming*, vol. 8, pp. 179–206, 1975.Google Scholar - M.D. McDonnell, H.P. Possingham, I.R. Ball, and E.A. Cousins “Mathematical methods for spatially cohesive reserve design,”
*Environmental Modeling and Assessment*, vol. 7, pp. 107–114, 2002.CrossRefGoogle Scholar - D.J. Nalle, J.L. Arthur, and J. Sessions “Designing compact and contiguous reserve networks with a hybrid heuristic approach,”
*Forest Science*, vol. 48, pp. 59–68, 2002.Google Scholar - A.O. Nicholls and C.R. Margules “An upgraded reserve selection algorithm,”
*Biological Conservation*, vol. 64, pp. 165–169, 1993.CrossRefGoogle Scholar - P. Nobili and A. Sassano, “Facets and lifting procedures for the set covering polytope,”
*Mathematical Programming*, vol. 45, pp. 111–137, 1989.Google Scholar - C. Van Nuffelen, “On the rank of the incidence matrix of a graph,”
*Cahiers du Centre d’ Etudes de Recherche Opérationnelle*,vol.15, pp.363–365, 1973.Google Scholar - H. önal and R.A. Briers, “Incorporating spatial criteria in optimum reserve network selection,” in
*Proceedings of the Royal Society*,*London B*, 2002, vol.269, pp.2437–2441.CrossRefGoogle Scholar - H. önal and R.A. Briers, “Selection of a minimum-boundary reserve network using integer programming,” in
*Proceedings of the Royal Society*,*London B*, 2003, vol.270, pp.1487–1491.CrossRefGoogle Scholar - H. Possingham, I. Ball, and S. Andelman, “Mathematical methods for identifying representative reserve networks,” in
*Quantitative Methods for Conservation Biology*, S. Ferson and M. Burgman (Eds.), Springer-Verlag: New York, 2000, pp.291–306.Google Scholar - R. Primack,
*Essentials of Conservation Biology*, Sinauer Associates, Sunderland, 2002.Google Scholar - M. S’nchez-Garcí a, M.I. Sobrón, and B. Victoriano, “On the set covering polytope: Facets with coefficients in {0,1,2,3},”
*Annals of Operations Research*, vol.81, pp.343–356, 1998.CrossRefGoogle Scholar - H. Sachs, “Über teiler, faktoren und charakteristische polynome von graphen. II,”
*Wiss Z. Tech. Hochsch Ilmenau*, vol.13,pp.405–412, 1967.Google Scholar - A. Sassano, “On the facial structure of the set covering polytope,”
*Mathematical Programming*, vol.44, pp.181–202, 1989.Google Scholar - P. Siitonen, A. Tanskanen, and A. Lehtinen, “Method for selection of old-forest reserves,”
*Conservation Biology*, vol.16, pp.1398–1408, 2002.CrossRefGoogle Scholar - P. Siitonen, A. Tanskanen, and A. Lehtinen, “Selecting forest reserves with a multiobjective spatial algorithm,”
*EnvironmentalScience & Policy*, vol.6, pp.301–309, 2003.Google Scholar - R.R. Vemuganti, “Applications of the set covering, set packing and partitioning models: A survey,” in
*Handbook of Combinatorial Optimization*, vol.1, D.-Z. Du and P. Pardalos (Eds.), Kluwer Academic Publishers: Boston, 1998, pp.573–746.Google Scholar - J.C. Williams, “Delineating protected wildlife corridors with multi-objective programming,”
*Environmental Modeling and Assessment*, vol.3, pp.77–86, 1998.CrossRefGoogle Scholar - L. Wolsey, “Faces for a linear inequality in 0-1 variables,”
*Mathematical Programming*, vol.8, pp.165–178, 1975.Google Scholar