1,590 research outputs found
Local Graph Coloring and Index Coding
We present a novel upper bound for the optimal index coding rate. Our bound
uses a graph theoretic quantity called the local chromatic number. We show how
a good local coloring can be used to create a good index code. The local
coloring is used as an alignment guide to assign index coding vectors from a
general position MDS code. We further show that a natural LP relaxation yields
an even stronger index code. Our bounds provably outperform the state of the
art on index coding but at most by a constant factor.Comment: 14 Pages, 3 Figures; A conference version submitted to ISIT 2013;
typos correcte
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
TDMA is Optimal for All-unicast DoF Region of TIM if and only if Topology is Chordal Bipartite
The main result of this work is that an orthogonal access scheme such as TDMA
achieves the all-unicast degrees of freedom (DoF) region of the topological
interference management (TIM) problem if and only if the network topology graph
is chordal bipartite, i.e., every cycle that can contain a chord, does contain
a chord. The all-unicast DoF region includes the DoF region for any arbitrary
choice of a unicast message set, so e.g., the results of Maleki and Jafar on
the optimality of orthogonal access for the sum-DoF of one-dimensional convex
networks are recovered as a special case. The result is also established for
the corresponding topological representation of the index coding problem
On topological relaxations of chromatic conjectures
There are several famous unsolved conjectures about the chromatic number that
were relaxed and already proven to hold for the fractional chromatic number. We
discuss similar relaxations for the topological lower bound(s) of the chromatic
number. In particular, we prove that such a relaxed version is true for the
Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of
Hadwiger from this point of view. For the latter, a similar statement was
already proven in an earlier paper of the first author with G. Tardos, our main
concern here is that the so-called odd Hadwiger conjecture looks much more
difficult in this respect. We prove that the statement of the odd Hadwiger
conjecture holds for large enough Kneser graphs and Schrijver graphs of any
fixed chromatic number
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