Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take any position in given uncertainty sets. Then, the cost function to be minimized is the sum of the distances for the worst positions of the vertices in their uncertainty sets. We propose two types of polynomial-time approximation algorithms. The first one relies on solving a deterministic counterpart of the problem where the uncertain distances are replaced with maximum pairwise distances. We study in details the resulting approximation ratio, which depends on the structure of the feasible subgraphs and whether the metric space is Ptolemaic or not. The second algorithm is a fully-polynomial time approximation scheme for the special case of $s-t$ paths.

翻译：许多组合优化问题可被表述为在满足特定性质的子图中搜索总权重最小的子图。本文假设顶点对应于度量空间中的点，且可在给定的不确定集合内任意移动。此时，待最小化的成本函数定义为顶点在其不确定集合内取最不利位置时的距离总和。我们提出两类多项式时间近似算法：第一类算法通过求解问题的确定性对应形式实现，其中不确定距离被替换为最大成对距离。我们详细研究了由此产生的近似比，该比值取决于可行子图的结构以及度量空间是否为托勒密空间。第二类算法是针对$s-t$路径特殊情形的完全多项式时间近似方案。