Generalized Interlinked Cycle Cover for Index Coding

TL;DR

This paper introduces generalized interlinked cycles (GIC) in directed graphs for index coding, providing an optimal scalar linear encoding scheme that outperforms existing methods for certain graph classes.

Contribution

It defines GIC structures as a generalization of cliques and cycles and develops an efficient, optimal encoding scheme exploiting these structures.

Findings

The proposed scheme is optimal for a class of digraphs.

It outperforms existing techniques like partial clique cover and local chromatic number.

The encoding scheme has linear time complexity.

Abstract

A source coding problem over a noiseless broadcast channel where the source is pre-informed about the contents of the cache of all receivers, is an index coding problem. Furthermore, if each message is requested by one receiver, then we call this an index coding problem with a unicast message setting. This problem can be represented by a directed graph. In this paper, we first define a structure (we call generalized interlinked cycles (GIC)) in directed graphs. A GIC consists of cycles which are interlinked in some manner (i.e., not disjoint), and it turns out that the GIC is a generalization of cliques and cycles. We then propose a simple scalar linear encoding scheme with linear time encoding complexity. This scheme exploits GICs in the digraph. We prove that our scheme is optimal for a class of digraphs with message packets of any length. Moreover, we show that our scheme can outperform existing techniques, e.g., partial clique cover, local chromatic number, composite-coding, and interlinked cycle cover.