Computational and Theoretical Challenges for Computing the Minimum Rank of a Graph

Abstract

The minimum rank of a graph G is the minimum of the ranks of all symmetric adjacency matrices of G. We present a new combinatorial bound for the minimum rank of an arbitrary graph G based on enumerating certain subsets of vertices of G satisfying matroid theoretic properties. We also present some computational and theoretical challenges associated with computing the minimum rank. This includes a conjecture that this bound on the minimum rank actually holds with equality for all graphs. History: This “Challenge” paper was invited by the Editor in Chief and based on the topics raised by the author at his plenary address at the 2022 INFORMS Computing Society Conference in Tampa, Florida. Funding: This work was supported by the National Science Foundation [Grant DMS-1720225]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1219 .