Approximate Capacity of Index Coding for Some Classes of Graphs

TL;DR

This paper establishes bounds on the approximation of the broadcast rate in index coding problems for certain graph classes, including planar and interval graphs, using clique covering schemes and graph theory insights.

Contribution

It provides a new approximation bound for the broadcast rate in index coding for graphs with bounded Ramsey numbers, extending to specific graph classes.

Findings

Clique covering scheme approximates broadcast rate within a factor depending on graph size.

For planar, line, and fuzzy circular interval graphs, approximation factor is at most n^{2/3}.

Theoretical bounds relate Ramsey numbers to index coding efficiency.

Abstract

For a class of graphs for which the Ramsey number $R(i,j)$ is upper bounded by $ci^aj^b$, for some constants $a,b,$ and $c$, it is shown that the clique covering scheme approximates the broadcast rate of every $n$-node index coding problem in the class within a multiplicative factor of $c^{\frac{1}{a+b+1}} n^{\frac{a+b}{a+b+1}}$ for every $n$. Using this theorem and some graph theoretic arguments, it is demonstrated that the broadcast rate of planar graphs, line graphs and fuzzy circular interval graphs is approximated by the clique covering scheme within a factor of $n^{\frac{2}{3}}$.