An integer program for positive semidefinite zero forcing in graphs

Abstract

Positive semidefinite (PSD) zero forcing is a dynamic graph process in which an initial subset of vertices are colored and may cause additional vertices to become colored through a set of color changing rules. Subsets which cause all other vertices to become colored are called PSD zero forcing sets; the PSD zero forcing number of a graph is the minimum cardinality attained by its PSD zero forcing sets. The PSD zero forcing number is of particular interest as it bounds solutions for the minimum rank and PSD min rank problems, both popular in linear algebra. This paper introduces blocking sets for PSD zero forcing sets which are used to formulate the first integer program (IP) for computing PSD zero forcing numbers of general graphs. It is shown that facets of the feasible region of this IP's linear relaxation correspond to zero forcing forts which induce connected subgraphs, but that identifying min cardinality connected forts is NP‐hard in general. Auxiliary IPs used to find these blocking sets are also given, enabling the master IP to be solved via constraint generation. Experiments comparing the proposed methods and existing algorithms are provided demonstrating improved runtime performance, particularly so in dense and sparse graphs.