We study set selection problems where the weights are uncertain. Instead of its exact weight, only an uncertainty interval containing its true weight is available for each element. In some cases, some solutions are universally optimal; i.e., they are optimal for every weight that lies within the uncertainty intervals. However, it may be that no universal optimal solution exists, unless we are revealed additional information on the precise values of some elements. In the minimum cost admissible query problem, we are tasked to (non-adaptively) find a minimum-cost subset of elements that, no matter how they are revealed, guarantee the existence of a universally optimal solution. We introduce thresholds under uncertainty to analyze problems of minimum cost admissible queries. Roughly speaking, for every element e, there is a threshold for its weight, below which e is included in all optimal solutions and a second threshold above which e is excluded from all optimal solutions. We show that computing thresholds and finding minimum cost admissible queries are essentially equivalent problems. Thus, the analysis of the minimum admissible query problem reduces to the problem of computing thresholds. We provide efficient algorithms for computing thresholds in the settings of minimum spanning trees, matroids, and matchings in trees; and NP-hardness results in the settings of s-t shortest paths and bipartite matching. By making use of the equivalence between the two problems these results translate into efficient algorithms for minimum cost admissible queries in the settings of minimum spanning trees, matroids, and matchings in trees; and NP-hardness results in the settings of s-t shortest paths and bipartite matching.

翻译：我们研究权重不确定的集合选择问题。对于每个元素，我们无法获知其精确权重，仅能获取包含真实权重的区间。在某些情况下，部分解具有全局最优性，即对于不确定性区间内的任意权重取值，这些解均为最优。然而若无额外信息揭示某些元素的精确值，可能不存在全局最优解。在最小代价可接受查询问题中，我们的目标是非自适应地找到一个最小代价元素子集，使得无论这些元素如何被揭示，都能保证全局最优解的存在。我们引入不确定性阈值来分析最小代价可接受查询问题。大致而言，每个元素e存在两个权重阈值：低于第一阈值时，所有最优解均包含e；高于第二阈值时，所有最优解均排除e。研究表明，计算阈值与求解最小代价可接受查询问题本质等价，因此后者可简化为阈值计算问题。针对最小生成树、拟阵和树匹配场景，我们提出高效阈值计算算法；针对s-t最短路径和二分图匹配场景，则给出NP困难性证明。利用两类问题的等价性，这些结论可转化为最小生成树、拟阵和树匹配场景中最小代价可接受查询的高效算法，以及s-t最短路径和二分图匹配场景的NP困难性结论。