Reducing the clique and chromatic number via edge contractions and vertex deletions

We consider the following problem: can a certain graph parameter of some given graph G be reduced by at least d, for some integer d, via at most k graph operations from some specified set S, for some given integer k? As graph parameters we take the chromatic number and the clique number. We let the set S consist of either an edge contraction or a vertex deletion. As all these problems are NP-complete for general graphs even if d is fixed, we restrict the input graph G to some special graph class. We continue a line of research that considers these problems for subclasses of perfect graphs, but our main results are full classifications, from a computational complexity point of view, for graph classes characterized by forbidding a single induced connected subgraph H

Survivability in hierarchical telecommunications networks

Abstract We consider the problem of designing a two level telecommunications network at minimum cost. The decisions involved are the locations of concentrators, the assignments of user nodes to concentrators and the installation of links connecting concentrators in a reliable backbone network. We define a reliable backbone network as one where there exist at least 2-edge disjoint paths between any pair of concentrator nodes. We formulate this problem as an integer program and propose a facial study of the associated polytope. We describe valid inequalities and give sufficient conditions for these inequalities to be facet defining. We also propose some reduction operations in order to speed up the separation procedures for these inequalities. Using these results, we devise a branch-and-cut algorithm and present some computational results

Vulnerability assessment of spatial networks: models and solutions

In this paper we present a collection of combinatorial optimization problems that allows to assess the vulnerability of spatial networks in the presence of disruptions. The proposed measures of vulnerability along with the model of failure are suitable in many applications where the consideration of failures in the transportation system is crucial. By means of computational results, we show how the proposed methodology allows us to find useful information regarding the capacity of a network to resist disruptions and under which circumstances the network collapses

Design of Survivable Networks: A survey

For the past few decades, combinatorial optimization techniques have been shown to be powerful tools for formulating and solving optimization problems arising from practical situations. In particular, many network design problems have been formulated as combinatorial optimization problems. With the advances of optical technologies and the explosive growth of the Internet, telecommunication networks have seen an important evolution and therefore, designing survivable networks has become a major objective for telecommunication operators. Over the past years, a big amount of research has then been done for devising efficient methods for survivable network models, and particularly cutting plane based algorithms. In this paper, we attempt to survey some of these models and the optimization methods used for solving them