Topological Additive Numbering of Directed Acyclic Graphs

We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let $D$ be a digraph and $f$ a labeling of its vertices with positive integers; denote by $S(v)$ the sum of labels over all neighbors of each vertex $v$. The labeling $f$ is called \emph{topological additive numbering} if $S(u) < S(v)$ for each arc $(u,v)$ of the digraph. The problem asks to find the minimum number $k$ for which $D$ has a topological additive numbering with labels belonging to $\{ 1, \ldots, k \}$, denoted by $\eta_t(D)$. We characterize when a digraph has topological additive numberings, give a lower bound for $\eta_t(D)$, and provide an integer programming formulation for our problem, characterizing when its coefficient matrix is totally unimodular. We also present some families for which $\eta_t(D)$ can be computed in polynomial time. Finally, we prove that this problem is \np-Hard even when its input is restricted to planar bipartite digraphs

An integer programming approach for the hyper-rectangular clustering problem with axis-parallel clusters and outliers

We present a mixed integer programming formulation for the problem of clustering a set of points in Rd with axis-parallel clusters, while allowing to discard a pre-specified number of points, thus declared to be outliers. We identify a family of valid inequalities separable in polynomial time, we prove that some inequalities from this family induce facets of the associated polytope, and we show that the dynamic addition of cuts coming from this family is effective in practice

A branch-and-cut algorithm for the routing and spectrum allocation problem

One of the most promising solutions to deal with huge data traffic demands in large communication networks is given by flexible optical networking, in particular the flexible grid (flexgrid) technology specified in the ITU-T standard G.694.1. In this specification, the frequency spectrum of an optical fiber link is divided into narrow frequency slots. Any sequence of consecutive slots can be used as a simple channel, and such a channel can be switched in the network nodes to create a lightpath. In this kind of networks, the problem of establishing lightpaths for a set of end-to-end demands that compete for spectrum resources is called the routing and spectrum allocation problem (RSA). Due to its relevance, RSA has been intensively studied in the last years. It has been shown to be NP-hard and different solution approaches have been proposed for this problem. In this paper we present several families of valid inequalities, valid equations, and optimality cuts for a natural integer programming formulation of RSA and, based on these results, we develop a branch-and-cut algorithm for this problem. Our computational experiments suggest that such an approach is effective at tackling this problem

Valid inequalities and complete characterizations of the 2-domination and P3-hull number polytope

Given a graph G = (V;E), a subset S V is 2-dominating if every vertex in S has at least two neighbors in S. The minimum cardinality of such a set is called the 2-domination number of G. Consider a process in discrete time that, starting with an initial set of marked vertices S, at each step marks all unmarked vertices having two marked neighbors. In such a process, the minimum number of initial vertices in S such that eventually all vertices are marked is called the P3-hull number of G. In this work, we explore a polyhedral relation between these two parameters and, in addition, we provide new families of valid inequalities for the associated polytopes. Finally, we give explicit descriptions of the polytopes associated to these problems when G is a path, a cycle, a complete graph, or a tree

Facet-generating procedures for the maximum-impact coloring polytope

Given two graphs G = (V, EG) and H = (V, EH) over the same set of vertices and given a set of colors C, the impact on H of a coloring c : V → C of G, denoted I(c), is the number of edges ij ∈ EH such that c(i) = c(j). In this setting, the maximum-impact coloring problem asks for a proper coloring c of G maximizing the impact I(c) on H. This problem naturally arises in the context of classroom allocation to courses, where it is desirable –but not mandatory– to assign lectures from the same course to the same classroom. In a previous work we identified several families of facet-inducing inequalities for a natural integer programming formulation of this problem. Most of these families were based on similar ideas, leading us to explore whether they can be expressed within a unified framework. In this work we tackle this issue, by presenting two procedures that construct valid inequalities from existing inequalities, based on extending individual colors to sets of colors and on extending edges of G to cliques in G, respectively. If the original inequality defines a facet and additional technical hypotheses are satisfied, then the obtained inequality also defines a facet. We show that these procedures can explain most of the inequalities presented in a previous work, we present a generic separation algorithm based on these procedures, and we report computational experiments showing that this approach is effective.Este documento es una versión del artículo publicado en Discrete Applied Mathematics 323, 96-112

The maximum 2D subarray polytope: facet-inducing inequalities and polyhedral computations

Given a matrix with real-valued entries, the maximum 2D subarray problem consists in finding a rectangular submatrix with consecutive rows and columns maximizing the sum of its entries. In this work we start a polyhedral study of an integer programming formulation for this problem.We thus define the 2D subarray polytope, explore conditions ensuring the validity of linear inequalities, and provide several families of facet-inducing inequalities. We also report com- putational experiments assessing the reduction of the dual bound for the linear relaxation achieved by these families of inequalities.Este documento es una versión del artículo publicado en Applied Mathematics 323, 286-301

Polyhedral studies on vertex coloring problems

Many variants of the vertex coloring problem have been de ned, such as precoloring extension, μ-coloring, (γ ; μ)-coloring, and list coloring. These problems are NP-hard, as they generalize the classical vertex coloring problem. On the other side, there exist several families of graphs for which some of these problems can be solved in polynomial time. The standard integer programming model for coloring problems uses a binary variable xvc for each vertex v and each color c to indicate whether v is assigned c or not. An extension of this model considers binary variables wc for each color c to indicate whether color c is used or not. In this work we study this formulation for the polynomial cases of the coloring problems mentioned above. In particular, we prove that if the classical vertex coloring problem yields an integer polytope for a family of graphs, then the same holds for μ-coloring, ( γ; μ)-coloring, and list coloring over the same family. We prove that the polytope associated to these problems over trees is integer and that adding the clique inequalities, the resulting polytope is integer over block graphs also. Finally, we give a new family of facet-inducing valid inequalities for the standard formulation and we provide empirical evidence suggesting that this family completely describes the polytope associated to these problems over cycles (and probably cactii graphs).Sociedad Argentina de Informática e Investigación Operativ