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

The min-rank of a graph was introduced by Haemers (Algebr. Methods Graph Theory 25:267–272, 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. (in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 197–206, 2006). In their paper, it was shown that the min-rank of a graph $\mathcal {G}$ characterizes the optimal scalar linear solution of an instance of the Index Coding with Side Information (ICSI) problem described by the graph $\mathcal {G}$.It was shown by Peeters (Combinatorica 16(3):417–431, 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.