425 research outputs found
Graph Theory versus Minimum Rank for Index Coding
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
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 -element subsets of as vertices and all pairs of
disjoint sets as edges, has chromatic number . 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 -spaces with -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
We investigate the completely positive semidefinite cone ,
a new matrix cone consisting of all 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 and , which are roughly
obtained by allowing variables to be positive semidefinite matrices instead of
scalars in the programs defining the classical parameters. We can
formulate these quantum parameters as conic linear programs over the cone
. 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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