On the Relation Between the Index Coding and the Network Coding Problems

TL;DR

This paper demonstrates that the Index Coding problem encapsulates key properties of the broader Network Coding problem, including limitations of linear coding schemes, through an efficient reduction.

Contribution

It establishes a reduction from Network Coding to Index Coding, showing that properties like the insufficiency of linear codes carry over.

Findings

Index Coding captures essential properties of Network Coding.

Linear codes are insufficient for optimal solutions in both problems.

The reduction preserves key coding limitations across problems.

Abstract

In this paper we show that the Index Coding problem captures several important properties of the more general Network Coding problem. An instance of the Index Coding problem includes a server that holds a set of information messages $X=\{x_1,...,x_k\}$ and a set of receivers $R$. Each receiver has some side information, known to the server, represented by a subset of $X$ and demands another subset of $X$. The server uses a noiseless communication channel to broadcast encodings of messages in $X$ to satisfy the receivers' demands. The goal of the server is to find an encoding scheme that requires the minimum number of transmissions. We show that any instance of the Network Coding problem can be efficiently reduced to an instance of the Index Coding problem. Our reduction shows that several important properties of the Network Coding problem carry over to the Index Coding problem. In particular, we prove that both scalar linear and vector linear codes are insufficient for achieving the minimal number of transmissions.