The edit distance between two graphs is a widely used measure of similarity that evaluates the smallest number of vertex and edge deletions/insertions required to transform one graph to another. It is NP-hard to compute in general, and a large number of heuristics have been proposed for approximating this quantity. With few exceptions, these methods generally provide upper bounds on the edit distance between two graphs. In this paper, we propose a new family of computationally tractable convex relaxations for obtaining lower bounds on graph edit distance. These relaxations can be tailored to the structural properties of the particular graphs via convex graph invariants. Specific examples that we highlight in this paper include constraints on the graph spectrum as well as (tractable approximations of) the stability number and the maximum-cut values of graphs. We prove under suitable conditions that our relaxations are tight (i.e., exactly compute the graph edit distance) when one of the graphs consists of few eigenvalues. We also validate the utility of our framework on synthetic problems as well as real applications involving molecular structure comparison problems in chemistry.

中文翻译：

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图编辑距离的凸图不变松弛

两个图之间的编辑距离是一种广泛使用的相似性度量，用于评估将一个图转换为另一个图所需的最少顶点和边删除/插入次数。一般来说，计算是 NP 难的，并且已经提出了大量的启发式方法来逼近这个数量。除了少数例外，这些方法通常提供两个图形之间编辑距离的上限。在本文中，我们提出了一个新的计算上易于处理的凸松弛系列，用于获得图编辑距离的下界。这些松弛可以通过凸图不变量来适应特定图的结构特性。我们在本文中强调的具体示例包括对图谱的约束以及图的稳定性数和最大割值（的易处理近似值）。我们证明在合适的条件下，当其中一个图包含很少的特征值时，我们的松弛是严格的（即，精确计算图编辑距离）。我们还验证了我们的框架在合成问题以及涉及化学中分子结构比较问题的实际应用中的实用性。