425 research outputs found

    Graph Theory versus Minimum Rank for Index Coding

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    We obtain novel index coding schemes and show that they provably outperform all previously known graph theoretic bounds proposed so far. Further, we establish a rather strong negative result: all known graph theoretic bounds are within a logarithmic factor from the chromatic number. This is in striking contrast to minrank since prior work has shown that it can outperform the chromatic number by a polynomial factor in some cases. The conclusion is that all known graph theoretic bounds are not much stronger than the chromatic number.Comment: 8 pages, 2 figures. Submitted to ISIT 201

    Topological lower bounds for the chromatic number: A hierarchy

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    This paper is a study of ``topological'' lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lov\'asz in 1978, in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology. This conjecture stated that the \emph{Kneser graph} \KG_{m,n}, the graph with all kk-element subsets of {1,2,...,n}\{1,2,...,n\} as vertices and all pairs of disjoint sets as edges, has chromatic number n−2k+2n-2k+2. Several other proofs have since been published (by B\'ar\'any, Schrijver, Dolnikov, Sarkaria, Kriz, Greene, and others), all of them based on some version of the Borsuk--Ulam theorem, but otherwise quite different. Each can be extended to yield some lower bound on the chromatic number of an arbitrary graph. (Indeed, we observe that \emph{every} finite graph may be represented as a generalized Kneser graph, to which the above bounds apply.) We show that these bounds are almost linearly ordered by strength, the strongest one being essentially Lov\'asz' original bound in terms of a neighborhood complex. We also present and compare various definitions of a \emph{box complex} of a graph (developing ideas of Alon, Frankl, and Lov\'asz and of \kriz). A suitable box complex is equivalent to Lov\'asz' complex, but the construction is simpler and functorial, mapping graphs with homomorphisms to Z2\Z_2-spaces with Z2\Z_2-maps.Comment: 16 pages, 1 figure. Jahresbericht der DMV, to appea

    Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone

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    We investigate the completely positive semidefinite cone CS+n\mathcal{CS}_+^n, a new matrix cone consisting of all n×nn\times n matrices that admit a Gram representation by positive semidefinite matrices (of any size). In particular we study relationships between this cone and the completely positive and doubly nonnegative cones, and between its dual cone and trace positive non-commutative polynomials. We use this new cone to model quantum analogues of the classical independence and chromatic graph parameters α(G)\alpha(G) and χ(G)\chi(G), which are roughly obtained by allowing variables to be positive semidefinite matrices instead of 0/10/1 scalars in the programs defining the classical parameters. We can formulate these quantum parameters as conic linear programs over the cone CS+n\mathcal{CS}_+^n. Using this conic approach we can recover the bounds in terms of the theta number and define further approximations by exploiting the link to trace positive polynomials.Comment: Fixed some typo
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