Local graph coloring and index coding

Abstract

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.

Topological Interference Management With Decoded Message Passing

“…Similarly to the equivalence between interference alignment and local coloring shown in [23], it follows immediately that one-to-one interference alignment with message passing is equivalent to fractional local acyclic set coloring. Thus, we have a new inner bound for the symmetric DoF due to interference alignment.…”

“…Accordingly, the required vector dimension T depends merely on the number of different colors in the in-neighborhood of the directed conflict graph. Thus, a feasible one-to-one interference alignment scheme under the TIM setting is equivalent to a proper local coloring on the directed conflict graph [23].…”

Some new results on index coding when the number of data is less than the number of receivers

“…Here, each receiver r i requires X[f (i)] := {x j |j ∈ f (i)}, where f (i) ⊂ [n] := {1, · · · , n} is the index set of the required data for receiver i, and has side information X[N (i)] := {x j |j ∈ N (i)}, where N (i) ⊂ [n] is the index set for the side information for receiver i. The function E(X, {N (·)}) : (0, 1) n → (0, 1) l is an index code of length l if E(X, {N (·)}) is decodable in the Mostly, the index coding problem has been considered in the case in which n = m and f (1), · · · , f(m) is a partition of [n] with |f (i)| = 1, ∀ i [1], [3]- [5]. (We refer to this case as C 1 .)…”

“…Thus, M (P 1 (G cl (I EX ))) has * in the positions (1, 4), (1,5), (4, 1), (4, 5), (5, 1) and (5, 4) in (2). Since N (2) = {1}, receiver r 2 knows…”

“…This implies that 2014 IEEE International Symposium on Information Theory M (P 1 (G cl (I EX ))) has 0 in the positions (3, 1), (3,4) and (3,5) and has * in the position (3,2). Similarly, we can incorporate N (1), N (4) and N (5).…”