AI 中文总结
本文探究度量修复问题的结构可处理性，为串并联图、有界树宽图给出伪多项式时间算法，证明其在路径宽≤6图、网格图上分别为弱、强NP难，还得到长度受限多割问题的新算法。
AI 中文摘要
给定每条边都标注正距离的图$G$，最少需要修改多少条边的距离才能使$G$成为一个度量图？已知该度量修复问题在一般图上是NP难的，现有研究主要聚焦于近似算法和针对输入距离函数属性的固定参数可处理性。本文探究图自身的哪些结构属性会使度量修复问题变得可处理。在正向结果方面，我们为串并联图给出了伪多项式时间算法，并通过推广得到了有界树宽图的对应算法。该结果的直接推论是得到了长度受限多割问题的新算法，其运行时间以适度扩充后图的树宽为参数。令人惊讶的是，伪多项式时间竟是最优可能的结果：我们补充证明，即使在路径宽最多为6的图上，度量修复问题仍是弱NP难的。我们还证明平面性也无济于事，该问题即使在网格图上仍然是强NP难的。
英文摘要
Given a graph $G$ labeled with positive distances on each edge, what is the fewest number of edge distances that must be modified for $G$ to become a metric? It is known that this metric repair problem is $\mathrm{NP}$-hard on general graphs, with prior work focusing on approximations and fixed-parameter tractability with respect to properties of the input distance function. In this paper, we ask what structural properties of the graph itself make metric repair tractable. On the positive side, we give pseudo-polynomial time algorithms for series-parallel graphs, and by generalization, graphs of bounded treewidth. An immediate consequence of this result is a new algorithm for the length-bounded multicut problem, with a parameterized runtime bound in terms of the treewidth of a modestly augmented graph. Surprisingly, pseudo-polynomial time turns out to be the best one can hope for: We complement our algorithm with a proof that metric repair is weakly $\mathrm{NP}$-hard even on graphs of pathwidth at most six. We also prove that planarity does not help either, as the problem remains strongly $\mathrm{NP}$-hard even on grid graphs.
