受限图类中图形编辑距离的复杂性
On the Complexity of Graph Edit Distance in Restricted Graph Classes
AI总结：
研究图形编辑距离在受限图类中的复杂性，通过回顾文献中的多项式时间归约明确其与子图同构等问题的关系，建立特定条件下最大公共边子图与图形编辑距离的对应关系，揭示无标签和带标签图不同的复杂性情况。
AI中文摘要：
图形编辑距离推广了几个著名的NP难问题，本身也是NP难的。然而，所考虑的图类、编辑成本函数和由此产生的计算复杂性之间的关系尚不清楚。我们通过回顾文献中的多项式时间归约来研究这种相互作用，这些归约将子图同构和最大公共诱导子图归约为图形编辑距离。对于这些经典问题，已知NP难和多项式时间可解情况之间有明显区别，我们明确了对图形编辑距离复杂性的影响。在特定成本函数下，我们在带标签和无标签图中建立了最大公共边子图和图形编辑距离之间的图类保持对应关系。在无标签设置中，当一个图是路径而另一个是树时，最大公共边子图问题是多项式时间可解的。相比之下，对于带标签的图，我们证明即使两个图都是路径，最大公共边子图和图形编辑距离仍然是NP难的。
英文摘要：
The graph edit distance generalizes several well-known NP-hard problems and is therefore NP-hard itself. However, the relationship between the considered graph class, the edit cost function, and the resulting computational complexity is not well understood. We investigate this interplay by revisiting polynomial-time reductions from the literature, which reduce subgraph isomorphism and maximum common induced subgraph to the graph edit distance. For these classical problems, a sharp distinction between NP-hard and polynomial-time solvable cases is known, and we make the implications for the complexity of the graph edit distance explicit. We establish a graph-class-preserving correspondence between the maximum common edge subgraph and graph edit distance under a specific cost function, both in labeled and unlabeled graphs. In the unlabeled setting, the maximum common edge subgraph problem is polynomial-time solvable when one graph is a path and the other is a tree. In contrast, for labeled graphs, we prove that both the maximum common edge subgraph and the graph edit distance remain NP-hard, even when both graphs are paths.
