Improved Approximation Algorithms for Index Coding

TL;DR

This paper introduces improved polynomial-time approximation algorithms for the index coding problem, achieving tighter bounds for graphs, digraphs, and specific subclasses like quasi-line graphs, surpassing previous approximation factors.

Contribution

It presents a general method for approximating the index coding rate with better factors for various graph classes, including graphs, digraphs, and quasi-line graphs.

Findings

Approximation factor of O(n/log^2 n) for graphs.

Approximation factor of O(n/log n) for digraphs.

Polynomial-time algorithm with a factor of 2 for quasi-line graphs.

Abstract

The index coding problem is concerned with broadcasting encoded information to a collection of receivers in a way that enables each receiver to discover its required data based on its side information, which comprises the data required by some of the others. Given the side information map, represented by a graph in the symmetric case and by a digraph otherwise, the goal is to devise a coding scheme of minimum broadcast length. We present a general method for developing efficient algorithms for approximating the index coding rate for prescribed families of instances. As applications, we obtain polynomial-time algorithms that approximate the index coding rate of graphs and digraphs on $n$ vertices to within factors of $O(n/\log^2 n)$ and $O(n/\log n)$ respectively. This improves on the approximation factors of $O(n/\log n)$ for graphs and $O(n \cdot \log \log n/\log n)$ for digraphs achieved by Blasiak, Kleinberg, and Lubetzky (IEEE Trans. Inform. Theory, 2013). For the family of quasi-line graphs, we exhibit a polynomial-time algorithm that approximates the index coding rate to within a factor of $2$. This improves on the approximation factor of $O(n^{2/3})$ achieved by Arbabjolfaei and Kim (ISIT, 2016) for graphs on $n$ vertices taken from certain sub-families of quasi-line graphs. Our approach is applicable for approximating a variety of additional graph and digraph quantities to within the same approximation factors. Specifically, it captures every graph quantity sandwiched between the independence number and the clique cover number and every digraph quantity sandwiched between the maximum size of an acyclic induced sub-digraph and the directed clique cover number.