541 research outputs found

    Boxicity and separation dimension

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    A family F\mathcal{F} of permutations of the vertices of a hypergraph HH is called 'pairwise suitable' for HH if, for every pair of disjoint edges in HH, there exists a permutation in F\mathcal{F} in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for HH is called the 'separation dimension' of HH and is denoted by Ļ€(H)\pi(H). Equivalently, Ļ€(H)\pi(H) is the smallest natural number kk so that the vertices of HH can be embedded in Rk\mathbb{R}^k such that any two disjoint edges of HH can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph HH is equal to the 'boxicity' of the line graph of HH. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to WG-2014. Some results proved in this paper are also present in arXiv:1212.6756. arXiv admin note: substantial text overlap with arXiv:1212.675

    Uniform hypergraphs containing no grids

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    A hypergraph is called an rƗr grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that Aiāˆ©Aj=Biāˆ©Bj=Ļ† for 1ā‰¤i<jā‰¤r and {pipe}Aiāˆ©Bj{pipe}=1 for 1ā‰¤i, jā‰¤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1āˆ©C2{pipe}={pipe}C2āˆ©C3{pipe}={pipe}C3āˆ©C1{pipe}=1, C1āˆ©C2ā‰ C1āˆ©C3. A hypergraph is linear, if {pipe}Eāˆ©F{pipe}ā‰¤1 holds for every pair of edges Eā‰ F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For rā‰„. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. Ā© 2013 Elsevier Ltd

    Non-Uniform Robust Network Design in Planar Graphs

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    Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every resource is equally vulnerable, and that the set of scenarios is implicitly given by a single budget constraint. This paper studies a robustness model of a different kind. We focus on \textbf{bulk-robustness}, a model recently introduced~\cite{bulk} for addressing the need to model non-uniform failure patterns in systems. We significantly extend the techniques used in~\cite{bulk} to design approximation algorithm for bulk-robust network design problems in planar graphs. Our techniques use an augmentation framework, combined with linear programming (LP) rounding that depends on a planar embedding of the input graph. A connection to cut covering problems and the dominating set problem in circle graphs is established. Our methods use few of the specifics of bulk-robust optimization, hence it is conceivable that they can be adapted to solve other robust network design problems.Comment: 17 pages, 2 figure

    On the extension complexity of combinatorial polytopes

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    In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching. We then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors. Using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs, including those used in quantum information and suspensions of cubic planar graphs.Comment: 15 pages, 3 figures, 2 table
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