90 research outputs found

    Facet-generating procedures for the maximum-impact coloring polytope

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    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

    Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

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    The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.Comment: 28 pages, survey pape

    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

    Topics in Packing and Scheduling

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    Packing and scheduling models include some of the most fundamental problems in operations research and computer science. These broad classes include a wide range of models with applications including logistics, production planning, wireless network design, circuit design, and cloud computing, to name a few. In this thesis we study three such models: dynamic node packing, interval scheduling with economies of scale, and temporal bin packing with half-capacity jobs; each extends on a well-known problem in packing and scheduling. While the problems are generally distinct, this research was broadly inspired by applications to cloud computing. Specifically, this thesis is motivated by problems cloud service providers face when servicing requests for virtual machines. In Chapter 2, we propose a dynamic version of the node packing problem. In this model, instead of being given the edges upfront, we model them as Bernoulli random variables. At each step, the decision maker selects an available node and then observes edges adjacent to this node. The goal is a policy that maximizes the expected value of the resulting packing. We model the problem as a Markov decision problem and conduct a polyhedral study of the problem's achievable probabilities polytope. We develop a variety of valid inequalities based on paths, cycles, and cliques. In Chapter 3, we study interval scheduling problems exhibiting economies of scale. An instance is given by a set of interval jobs and a cost function. Specifically, we focus on the max-weight function and non-negative, non-decreasing concave functions of total schedule weight. The goal is a partition of the jobs minimizing the total cost with the constraint that jobs within the same schedule cannot overlap. We propose a set covering formulation and a column generation algorithm to solve its linear relaxation, providing efficient pricing algorithms for the studied cases. To obtain integer solutions, we extend the column generation approach using branch-and-price. In Chapter 4, we study a different model with interval jobs. In this problem, interval jobs are partitioned into bins such that at most two jobs in a bin overlap at once. The decision maker is tasked with minimizing the time-average number of bins required to pack all jobs. We call this problem temporal bin packing with half-capacity jobs; it is a special case of the general temporal bin packing problem with bounded parallelism. We study the worst-case performance of a well-known static lower bound, and, motivated by this analysis, we introduce a novel lower bound and integer programming formulation based on formulating the problem as a series of matching problems. We provide theoretical guarantees on the relative strengths of the static bound, the matching-based bound, and various linear programming bounds.Ph.D

    Reformulation and decomposition of integer programs

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    In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

    Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

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    International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

    Q(sqrt(-3))-Integral Points on a Mordell Curve

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    We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4
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