On the Capacity Region for Index Coding

A new inner bound on the capacity region of a general index coding problem is established. Unlike most existing bounds that are based on graph theoretic or algebraic tools, the bound is built on a random coding scheme and optimal decoding, and has a simple polymatroidal single-letter expression. The utility of the inner bound is demonstrated by examples that include the capacity region for all index coding problems with up to five messages (there are 9846 nonisomorphic ones).Comment: 5 pages, 6 figures, accepted to the 2013 IEEE International Symposium on Information Theory (ISIT), Istanbul, Turkey, July 201

On Index Coding and Graph Homomorphism

In this work, we study the problem of index coding from graph homomorphism perspective. We show that the minimum broadcast rate of an index coding problem for different variations of the problem such as non-linear, scalar, and vector index code, can be upper bounded by the minimum broadcast rate of another index coding problem when there exists a homomorphism from the complement of the side information graph of the first problem to that of the second problem. As a result, we show that several upper bounds on scalar and vector index code problem are special cases of one of our main theorems. For the linear scalar index coding problem, it has been shown in [1] that the binary linear index of a graph is equal to a graph theoretical parameter called minrank of the graph. For undirected graphs, in [2] it is shown that $\mathrm{minrank}(G) = k$ if and only if there exists a homomorphism from $\bar{G}$ to a predefined graph $\bar{G}_k$. Combining these two results, it follows that for undirected graphs, all the digraphs with linear index of at most k coincide with the graphs $G$ for which there exists a homomorphism from $\bar{G}$ to $\bar{G}_k$. In this paper, we give a direct proof to this result that works for digraphs as well. We show how to use this classification result to generate lower bounds on scalar and vector index. In particular, we provide a lower bound for the scalar index of a digraph in terms of the chromatic number of its complement. Using our framework, we show that by changing the field size, linear index of a digraph can be at most increased by a factor that is independent from the number of the nodes.Comment: 5 pages, to appear in "IEEE Information Theory Workshop", 201

Optimality of Orthogonal Access for One-dimensional Convex Cellular Networks

It is shown that a greedy orthogonal access scheme achieves the sum degrees of freedom of all one-dimensional (all nodes placed along a straight line) convex cellular networks (where cells are convex regions) when no channel knowledge is available at the transmitters except the knowledge of the network topology. In general, optimality of orthogonal access holds neither for two-dimensional convex cellular networks nor for one-dimensional non-convex cellular networks, thus revealing a fundamental limitation that exists only when both one-dimensional and convex properties are simultaneously enforced, as is common in canonical information theoretic models for studying cellular networks. The result also establishes the capacity of the corresponding class of index coding problems

Index Coding: Rank-Invariant Extensions

An index coding (IC) problem consisting of a server and multiple receivers with different side-information and demand sets can be equivalently represented using a fitting matrix. A scalar linear index code to a given IC problem is a matrix representing the transmitted linear combinations of the message symbols. The length of an index code is then the number of transmissions (or equivalently, the number of rows in the index code). An IC problem ${\cal I}_{ext}$ is called an extension of another IC problem ${\cal I}$ if the fitting matrix of ${\cal I}$ is a submatrix of the fitting matrix of ${\cal I}_{ext}$. We first present a straightforward $m$\textit{-order} extension ${\cal I}_{ext}$ of an IC problem ${\cal I}$ for which an index code is obtained by concatenating $m$ copies of an index code of ${\cal I}$. The length of the codes is the same for both ${\cal I}$ and ${\cal I}_{ext}$, and if the index code for ${\cal I}$ has optimal length then so does the extended code for ${\cal I}_{ext}$. More generally, an extended IC problem of ${\cal I}$ having the same optimal length as ${\cal I}$ is said to be a \textit{rank-invariant} extension of ${\cal I}$. We then focus on $2$-order rank-invariant extensions of ${\cal I}$, and present constructions of such extensions based on involutory permutation matrices

A New Class of Index Coding Instances Where Linear Coding is Optimal

We study index-coding problems (one sender broadcasting messages to multiple receivers) where each message is requested by one receiver, and each receiver may know some messages a priori. This type of index-coding problems can be fully described by directed graphs. The aim is to find the minimum codelength that the sender needs to transmit in order to simultaneously satisfy all receivers' requests. For any directed graph, we show that if a maximum acyclic induced subgraph (MAIS) is obtained by removing two or fewer vertices from the graph, then the minimum codelength (i.e., the solution to the index-coding problem) equals the number of vertices in the MAIS, and linear codes are optimal for this index-coding problem. Our result increases the set of index-coding problems for which linear index codes are proven to be optimal.Comment: accepted and to be presented at the 2014 International Symposium on Network Coding (NetCod

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

A class of index coding problems with rate 1/3

An index coding problem with $n$ messages has symmetric rate $R$ if all $n$ messages can be conveyed at rate $R$. In a recent work, a class of index coding problems for which symmetric rate $\frac{1}{3}$ is achievable was characterised using special properties of the side-information available at the receivers. In this paper, we show a larger class of index coding problems (which includes the previous class of problems) for which symmetric rate $\frac{1}{3}$ is achievable. In the process, we also obtain a stricter necessary condition for rate $\frac{1}{3}$ feasibility than what is known in literature.Comment: Shorter version submitted to ISIT 201

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