Improved Approximation Algorithms for Index Coding

TL;DR

The paper studies index coding with side information, where the rate $\beta(G)$ is bounded by the clique-cover number and the MAIS bound. It introduces a general, Ramsey-theory–driven framework that reduces approximating $\beta(G)$ to finding economical clique covers relative to $\mathrm{MAIS}(G)$, via bounds of the form $\mathrm{CC}(G) \le (\sum 1/f(i)) \mathrm{MAIS}(G)$. For broad graph/digraph families, the framework yields polynomial-time algorithms achieving $O\left(\frac{n}{\log^2 n}\right)$ (graphs) and $O\left(\frac{n}{\log n}\right)$ (digraphs) factors, with a tight 2-approximation for quasi-line graphs; the results extend to multiple index-coding measures and to generalized index coding through MES. The approach unifies and improves prior results (e.g., Blasiak–Kleinberg–Lubetzky) and provides new algorithmic guarantees across hereditary graph families, guided by Ramsey numbers and forbidden-structure decompositions. Overall, the work advances practical approximations for index coding and related graph-digraph quantities, with implications for network coding and related combinatorial problems.

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.

Theorems & Definitions (16)

• Theorem 1.1: BoppanaH92
• Definition 1.2: Ramsey Numbers
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• Lemma 2.2