Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques

TL;DR

This paper introduces interlinked-cycle structures in graphs to improve index coding, generalizing previous cycle and clique methods, and demonstrates the scheme's optimality and superiority over existing approaches for certain graph classes.

Contribution

It defines interlinked-cycle structures and the ICC scheme, extending index coding techniques beyond cycles and cliques, and proves their optimality for specific graph classes.

Findings

ICC scheme is optimal for an infinite class of digraphs.

ICC outperforms existing graph-based index coding schemes.

Scalar linear index codes are proven optimal for certain graphs.

Abstract

We consider a graphical approach to index coding. While cycles have been shown to provide coding gain, only disjoint cycles and cliques (a specific type of overlapping cycles) have been exploited in existing literature. In this paper, we define a more general form of overlapping cycles, called the interlinked-cycle (IC) structure, that generalizes cycles and cliques. We propose a scheme, called the interlinked-cycle-cover (ICC) scheme, that leverages IC structures in digraphs to construct scalar linear index codes. We characterize a class of infinitely many digraphs where our proposed scheme is optimal over all linear and non-linear index codes. Consequently, for this class of digraphs, we indirectly prove that scalar linear index codes are optimal. Furthermore, we show that the ICC scheme can outperform all existing graph-based schemes (including partial-clique-cover and fractional-local-chromatic number schemes), and a random-coding scheme (namely, composite coding) for certain graphs.