Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures

TL;DR

This paper introduces a polynomial time algorithm to compute the min-rank of graphs with simple tree structures, expanding the class of graphs with efficiently computable min-ranks relevant to index coding.

Contribution

It presents a new family of graphs with efficiently computable min-ranks and a dynamic programming algorithm for their computation, along with a recognition method.

Findings

Polynomial time algorithm for min-rank of simple tree structure graphs

Recognition algorithm for these graph families

Extension of polynomial-time computability to new graph classes

Abstract

The min-rank of a graph was introduced by Haemers (1978) to bound the Shannon capacity of a graph. This parameter of a graph has recently gained much more attention from the research community after the work of Bar-Yossef et al. (2006). In their paper, it was shown that the min-rank of a graph G characterizes the optimal scalar linear solution of an instance of the Index Coding with Side Information (ICSI) problem described by the graph G. It was shown by Peeters (1996) that computing the min-rank of a general graph is an NP-hard problem. There are very few known families of graphs whose min-ranks can be found in polynomial time. In this work, we introduce a new family of graphs with efficiently computed min-ranks. Specifically, we establish a polynomial time dynamic programming algorithm to compute the min-ranks of graphs having simple tree structures. Intuitively, such graphs are obtained by gluing together, in a tree-like structure, any set of graphs for which the min-ranks can be determined in polynomial time. A polynomial time algorithm to recognize such graphs is also proposed.