On Critical Index Coding Problems

TL;DR

This paper investigates conditions under which side information can be removed in index coding without affecting capacity, introducing the concept of unicycles to identify critical edges and connecting it to existing bounds.

Contribution

It introduces the unicycle concept for side information graphs and establishes criticality conditions, linking them to known capacity bounds in index coding.

Findings

Edges in unicycles are critical and cannot be removed without reducing capacity

Partial converse relates unicycles to maximal acyclic subgraph bounds

Provides new insights into the structure of critical side information in index coding

Abstract

The question of under what condition some side information for index coding can be removed without affecting the capacity region is studied, which was originally posed by Tahmasbi, Shahrasbi, and Gohari. To answer this question, the notion of unicycle for the side information graph is introduced and it is shown that any edge that belongs to a unicycle is critical, namely, it cannot be removed without reducing the capacity region. Although this sufficient condition for criticality is not necessary in general, a partial converse is established, which elucidates the connection between the notion of unicycle and the maximal acylic induced subgraph outer bound on the capacity region by Bar-Yossef, Birk, Jayram, and Kol.