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 ‐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.

中文翻译：

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用于图中正半定零强迫的整数程序

正半定（PSD）零强制是一种动态图过程，在该过程中，顶点的初始子集被着色，并可能通过一组颜色变化规则使其他顶点变为彩色。使所有其他顶点变为彩色的子集称为PSD零强制集；图的PSD迫零数是其PSD迫零集获得的最小基数。PSD零逼迫数特别受关注，因为它限制了在线性代数中都流行的最小秩和PSD最小秩问题的解。本文介绍了PSD零强制集的阻塞集，这些阻塞集用于制定用于计算一般图的PSD零强制数的第一个整数程序（IP）。结果表明，该IP'可行区域的各个方面-一般来说很难。还提供了用于查找这些阻止集的辅助IP，从而可以通过约束生成来解决主IP。提供了将所提出的方法与现有算法进行比较的实验，证明了改进的运行时性能，尤其是在密集图和稀疏图中。