Index coding via linear programming

Abstract

Index Coding has received considerable attention recently motivated in part by applications such as fast video-on-demand and efficient communication in wireless networks and in part by its connection to Network Coding. Optimal encoding schemes and efficient heuristics were studied in various settings, while also leading to new results for Network Coding such as improved gaps between linear and non-linear capacity as well as hardness of approximation. The basic setting of Index Coding encodes the side-information relation, the problem input, as an undirected graph and the fundamental parameter is the broadcast rate β, the average communication cost per bit for sufficiently long messages (i.e. the non-linear vector capacity). Recent nontrivial bounds on β were derived from the study of other Index Coding capacities (e.g. the scalar capacity β 1 ) by Bar-Yossef et al (2006), Lubetzky and Stav (2007) and Alon et al (2008). However, these indirect bounds shed little light on the behavior of β: there was no known polynomial-time algorithm for approximating β in a general network to within a nontrivial (i.e. o(n)) factor, and the exact value of β remained unknown for any graph where Index Coding is nontrivial.Our main contribution is a direct information-theoretic analysis of the broadcast rate β using linear programs, in contrast to previous approaches that compared β with graph-theoretic parameters. This allows us to resolve the aforementioned two open questions. We provide a polynomial-time algorithm with a nontrivial approximation ratio for computing β in a general network along with a polynomial-time decision procedure for recognizing instances with β = 2. In addition, we pinpoint β precisely for various classes of graphs (e.g. for various Cayley graphs of cyclic groups) thereby simultaneously improving the previously known upper and lower bounds for these graphs. Via this approach we construct graphs where the difference between β and its trivial lower bound is linear in the number of vertices and ones where β is uniformly bounded while its upper bound derived from the naive encoding scheme is polynomially worse.

Interference, cooperation and connectivity — A degrees of freedom perspective

“…Second, the derivation of the outer bound itself follows a novel approach relative to prior work on related problems. While previously obtained DoF outer bounds in [2], [3], [5] follow from a common genie-aided multiple-access channel argument, the outer bound derived here fundamentally relies on the sub-modularity property of entropy functions, more commonly exploited in network coding converses [6], [7]. While the sub-modularity of entropy functions is an elementary property by itself, it is the manner in which this property is applied that is quite insightful.…”

“…The intuition that Lemma 1 should be applied in this manner comes from an interference alignment perspective, to be elaborated upon in the full paper. Finally, (7) follows from the observation that from Y n 1 , once all interference due to W 1 , W 4 is removed, we obtain an interference-free AWGN channel to transmitter 2, from which W 2 can be decoded subject to noise reduction operations that only have an o(log(SNR)) impact. Therefore (7).…”

“…Finally, (7) follows from the observation that from Y n 1 , once all interference due to W 1 , W 4 is removed, we obtain an interference-free AWGN channel to transmitter 2, from which W 2 can be decoded subject to noise reduction operations that only have an o(log(SNR)) impact. Therefore (7). What we have so far is the first set of bounds:…”

Index coding problem with side information repositories

“…The minimum broadcast rate is denoted by β. The general problem of computing minimum broadcast rate over all possible encoding functions is very hard to characterize [6].…”

“…Authors of [14] provide a polynomial time algorithm based on graph coloring for approximating minranks for undirected graph instances whose minrank is k. Recently, interference alignment approach has been applied to the problem of index coding [15]. In [6], a set of linear programs were developed whose optima effectively provide lower and upper bounds on the optimum broadcast rate β. Fractional coloring and coloring of the complement of the undirected side information graph are the most well known popular upper bounds to the optimum rate β [6] yielding achievable linear binary schemes.…”