Optimal Index Codes with Near-Extreme Rates

TL;DR

This paper characterizes graphs and digraphs with near-extreme min-ranks related to optimal scalar linear index codes, introduces a new upper bound, and explores computational complexity aspects of min-rank determination.

Contribution

It provides a characterization of near-extreme min-rank graphs and digraphs, introduces the circuit-packing bound, and analyzes the complexity of min-rank decision problems.

Findings

Min-rank two decision problem is NP-complete for digraphs.

Polynomial-time algorithms for certain digraph families.

Circuit-packing bound often tighter than previous bounds.

Abstract

The min-rank of a digraph was shown by Bar-Yossef et al. (2006) to represent the length of an optimal scalar linear solution of the corresponding instance of the Index Coding with Side Information (ICSI) problem. In this work, the graphs and digraphs of near-extreme min-ranks are characterized. Those graphs and digraphs correspond to the ICSI instances having near-extreme transmission rates when using optimal scalar linear index codes. In particular, it is shown that the decision problem whether a digraph has min-rank two is NP-complete. By contrast, the same question for graphs can be answered in polynomial time. Additionally, a new upper bound on the min-rank of a digraph, the circuit-packing bound, is presented. This bound is often tighter than the previously known bounds. By employing this new bound, we present several families of digraphs whose min-ranks can be found in polynomial time.