7 research outputs found

    Optimal k-fold colorings of webs and antiwebs

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    A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, ..., x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a k-fold x-coloring is the k-th chromatic number of G, denoted by \chi_k(G). We determine the exact value of this parameter when G is a web or an antiweb. Our results generalize the known corresponding results for odd cycles and imply necessary and sufficient conditions under which \chi_k(G) attains its lower and upper bounds based on the clique, the fractional chromatic and the chromatic numbers. Additionally, we extend the concept of \chi-critical graphs to \chi_k-critical graphs. We identify the webs and antiwebs having this property, for every integer k <= 1.Comment: A short version of this paper was presented at the Simp\'osio Brasileiro de Pesquisa Operacional, Brazil, 201

    Transitive Packing: A Unifying Concept in Combinatorial Optimization

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    This paper attempts to give a better understanding of the facial structure of previously separately investigated polyhedra. It introduces the notion of transitive packing and the transitive packing polytope. Polytopes that turn out to be special cases of the transitive packing polytope are, among others, the node packing polytope, the acyclic subdigraph polytope, the bipartite subgraph polytope, the planar subgraph polytope, the clique partitioning polytope, the partition polytope, the transitive acyclic subdigraph polytope, the interval order polytope, and the relatively transitive subgraph polytope. We give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope,in this way introducing generalized cycle, generalized clique, generalized antihole, generalized antiweb, and odd partition inequalities. These classes subsume several known classes of valid inequalities for several of the special cases and give also many new inequalities for several other special cases. For some of the classes we also prove a lower bound for their Gomory-Chvdtal rank. Finally, we relate the concept of transitive packing to generalized (set) packing and covering as well as to balanced and ideal matrices

    Generating facets for the independence system

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    In this paper, we present procedures to obtain facet-defining inequalities for the independence system polytope. These procedures are defined for inequalities which are not necessarily rank inequalities. We illustrate the use of these procedures by der iving strong valid inequalities for the acyclic induced subgraph, triangle free induced subgraph, bipartite induced subgraph, and knapsack polytopes. Finally, we derive a new family of facet-defining ineq ualities for the independence system polytope by adding a set of edges to antiwebs.© 2009 Society for Industrial and Applied Mathematics

    ProblĂšme de k-SĂ©parateur

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    Let G be a vertex-weighted undirected graph. We aim to compute a minimum weight subset of vertices whose removal leads to a graph where the size of each connected component is less than or equal to a given positive number k. If k = 1 we get the classical vertex cover problem. Many formulations are proposed for the problem. The linear relaxations of these formulations are theoretically compared. A polyhedral study is proposed (valid inequalities, facets, separation algorithms). It is shown that the problem can be solved in polynomial time for many special cases including the path, the cycle and the tree cases and also for graphs not containing some special induced sub-graphs. Some (k + 1)-approximation algorithms are also exhibited. Most of the algorithms are implemented and compared. The k-separator problem has many applications. If vertex weights are equal to 1, the size of a minimum k-separator can be used to evaluate the robustness of a graph or a network. Another application consists in partitioning a graph/network into different sub-graphs with respect to different criteria. For example, in the context of social networks, many approaches are proposed to detect communities. By solving a minimum k-separator problem, we get different connected components that may represent communities. The k-separator vertices represent persons making connections between communities. The k-separator problem can then be seen as a special partitioning/clustering graph problemConsidĂ©rons un graphe G = (V,E,w) non orientĂ© dont les sommets sont pondĂ©rĂ©s et un entier k. Le problĂšme Ă  Ă©tudier consiste Ă  la construction des algorithmes afin de dĂ©terminer le nombre minimum de nƓuds qu’il faut enlever au graphe G pour que toutes les composantes connexes restantes contiennent chacune au plus k-sommets. Ce problĂšme nous l’appelons problĂšme de k-SĂ©parateur et on dĂ©signe par k-sĂ©parateur le sous-ensemble recherchĂ©. Il est une gĂ©nĂ©ralisation du Vertex Cover qui correspond au cas k = 1 (nombre minimum de sommets intersectant toutes les arĂȘtes du graphe

    Integer and Constraint programming methods for mutually Orthogonal Latin Squares.

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    This thesis examines the Orthogonal Latin Squares (OLS) problem from the viewpoint of Integer and Constraint programming. An Integer Programming (IP) model is proposed and the associated polytope is analysed. We identify several families of strong valid inequalities, namely inequalities arising from cliques, odd holes, antiwebs and wheels of the associated intersection graph. The dimension of the OLS polytope is established and it is proved that certain valid inequalities are facet-inducing. This analysis reveals also a new family of facet-defining inequalities for the polytope associated with the Latin square problem. Separation algorithms of the lowest complexity are presented for particular families of valid inequalities. We illustrate a method for reducing problem's symmetry, which extends previously known results. This allows us to devise an alternative proof for the non-existence of an OLS structure for n = 6, based solely on Linear Programming. Moreover, we present a more general Branch & Cut algorithm for the OLS problem. The algorithm exploits problem structure via integer preprocessing and a specialised branching mechanism. It also incorporates families of strong valid inequalities. Computational analysis is conducted in order to illustrate the significant improvements over simple Branch & Bound. Next, the Constraint Programming (CP) paradigm is examined. Important aspects of designing an efficient CP solver, such as branching strategies and constraint propagation procedures, are evaluated by comprehensive, problem-specific, experiments. The CP algorithms lead to computationally favourable results. In particular, the infeasible case of n = 6, which requires enumerating the entire solution space, is solved in a few seconds. A broader aim of our research is to successfully integrate IP and CP. Hence, we present ideas concerning the unification of IP and CP methods in the form of hybrid algorithms. Two such algorithms are presented and their behaviour is analysed via experimentation. The main finding is that hybrid algorithms are clearly more efficient, as problem size grows, and exhibit a more robust performance than traditional IP and CP algorithms. A hybrid algorithm is also designed for the problem of finding triples of Mutually Orthogonal Latin Squares (MOLS). Given that the OLS problem is a special form of an assignment problem, the last part of the thesis considers multidimensional assignment problems. It introduces a model encompassing all assignment structures, including the case of MOLS. A necessary condition for the existence of an assignment structure is revealed. Relations among assignment problems are also examined, leading to a proposed hierarchy. Further, the polyhedral analysis presented unifies and generalises previous results

    Graph theoretic generalizations of clique: optimization and extensions

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    This dissertation considers graph theoretic generalizations of the maximum clique problem. Models that were originally proposed in social network analysis literature, are investigated from a mathematical programming perspective for the first time. A social network is usually represented by a graph, and cliques were the first models of "tightly knit groups" in social networks, referred to as cohesive subgroups. Cliques are idealized models and their overly restrictive nature motivated the development of clique relaxations that relax different aspects of a clique. Identifying large cohesive subgroups in social networks has traditionally been used in criminal network analysis to study organized crimes such as terrorism, narcotics and money laundering. More recent applications are in clustering and data mining wireless networks, biological networks as well as graph models of databases and the internet. This research has the potential to impact homeland security, bioinformatics, internet research and telecommunication industry among others. The focus of this dissertation is a degree-based relaxation called k-plex. A distance-based relaxation called k-clique and a diameter-based relaxation called k-club are also investigated in this dissertation. We present the first systematic study of the complexity aspects of these problems and application of mathematical programming techniques in solving them. Graph theoretic properties of the models are identified and used in the development of theory and algorithms. Optimization problems associated with the three models are formulated as binary integer programs and the properties of the associated polytopes are investigated. Facets and valid inequalities are identified based on combinatorial arguments. A branch-and-cut framework is designed and implemented to solve the optimization problems exactly. Specialized preprocessing techniques are developed that, in conjunction with the branch-and-cut algorithm, optimally solve the problems on real-life power law graphs, which is a general class of graphs that include social and biological networks. Computational experiments are performed to study the effectiveness of the proposed solution procedures on benchmark instances and real-life instances. The relationship of these models to the classical maximum clique problem is studied, leading to several interesting observations including a new compact integer programming formulation. We also prove new continuous non-linear formulations for the classical maximum independent set problem which maximize continuous functions over the unit hypercube, and characterize its local and global maxima. Finally, clustering and network design extensions of the clique relaxation models are explored
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