Graciela L. Nasini

A Generalization of the Perfect Graph Theorem Under the Disjunctive Index

In this paper, we relate antiblocker duality between polyhedra, graph theory, and the disjunctive procedure. In particular, we analyze the behavior of the disjunctive procedure over the clique relaxation, ( G), of the stable set polytope in a graph R G, and the one associated to its complementary graph, R( G). We obtain a generalization of the Perfect Graph Theorem, proving that the disjunctive indices of R( G) and R( G) always coincide.

Lift and project relaxations for the matching and related polytopes

We compare lift and project methods given by Lovasz and Schrijver (the N+ and N procedures) and by Balas, Ceria and Cornuejols (the disjunctive procedure) when working on the matching, perfect matching and covering polytopes. When the underlying graph is the complete graph of n=2s + 1 nodes we obtain that the disjunctive index for all problems is s2, the N+-index for the matching and perfect matching problems is s (extending a result by Stephen and Tuncel), the N-index for the perfect matching problem is s, and the N+ and N indices for the covering problem and the N-index for the matching problem are strictly greater than s.

A polyhedral approach for the equitable coloring problem

In this work we study the polytope associated with a 0,1-integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be facet-defining inequalities. We also present computational evidence that shows the efficacy of these inequalities used in a cutting-plane algorithm.

The packing coloring problem for (q,q-4) graphs

It is known that computing the packing chromatic number of a graph is an NP-hard problem, even when restricted to tree graphs. This fact naturally leads to the search of graph families where this problem is polynomial time solvable. Babel et al. (2001) showed that a large variety of NP-complete problems can be efficiently solved for the class of (q,q−4) graphs, for every fixed q. In this work we show that also to compute the packing chromatic number can be efficiently solved for the class of (q,q−4) graphs.

The k-limited packing and k-tuple domination problems in strongly chordal, P4-tidy and split graphs☆

Abstract The notion of k-limited packing in a graph is a generalization of 2-packing. For a given non negative integer k, a subset B of vertices is a k-limited packing if there are at most k elements of B in the closed neighborhood of every vertex. On the other side, a k-tuple domination set in a graph is a subset of vertices D such that every vertex has at least k elements of D in its closed neighborhood. In this work we first reveal a strong relationship between these notions, and obtain from a result due to Liao and Chang (2002), the polynomiality of the k-limited packing problem for strongly chordal graphs. We also prove that, in coincidence with the case of domination, the k-limited packing problem is NP-complete for split graphs. Finally, we prove that both problems are polynomial for the non-perfect class of P4-tidy graphs, including the perfect classes of P4-sparse graphs and cographs.

The disjunctive procedure and blocker duality

In this paper we relate two rather different branches of polyhedral theory in linear optimization problems: the blocking type polyhedra and the disjunctive procedure of Balas et al. For this purpose, we define a disjunctive procedure over blocking type polyhedra with vertices in [0, 1]n, study its properties, and analyze its behavior under blocker duality. We compare the indices of the procedure over a pair of blocking clutter polyhedra, obtaining that they coincide.

A DSATUR-based algorithm for the Equitable Coloring Problem

This paper describes a new exact algorithm for the Equitable Coloring Problem, a coloring problem where the sizes of two arbitrary color classes differ in at most one unit. Based on the well known DSatur algorithm for the classic Coloring Problem, a pruning criterion arising from equity constraints is proposed and analyzed. The good performance of the algorithm is shown through computational experiments over random and benchmark instances.

Some advances on the set covering polyhedron of circulant matrices

Studying the set covering polyhedron of consecutive ones circulant matrices, Argiroffo and Bianchi found a class of facet defining inequalities, induced by a particular family of circulant minors. In this work we extend these results to inequalities associated with every circulant minor. We also obtain polynomial separation algorithms for particular classes of such inequalities.

Lovász-Schrijver SDP-operator and a superclass of near-perfect graphs

Abstract We study the Lovasz-Schrijver SDP-operator applied to the fractional stable set polytope of graphs. The problem of obtaining a combinatorial characterization of graphs for which the SDP-operator generates the stable set polytope in one step has been open since 1990. In an earlier publication, we named these graphs N + -perfect. In the current contribution, we propose a conjecture on combinatorial characterization of N + -perfect graphs and make progress towards such a full combinatorial characterization by establishing a new, close relationship among N + -perfect graphs, near-bipartite graphs and a newly introduced concept of full-support-perfect graphs.

Near-perfect graphs with polyhedral N+(G)

Abstract One of the beautiful results due to Grotschel, Lovasz and Schrijver is the fact that the theta body of a graph G is polyhedral if and only if G is perfect. Related to the theta body of G is a foundational construction of an operator on polytopes, called N + ( ⋅ ) , by Lovasz and Schrijver. Here, we initiate the pursuit of a characterization theorem analogous to the one above by Grotschel, Lovasz and Schrijver, replacing the theta body of G by N + ( G ) and searching for the combinatorial counterpart to replace the class of perfect graphs.