Given subsets of uncertain values, we study the problem of identifying the subset of minimum total value (sum of the uncertain values) by querying as few values as possible. This set selection problem falls into the field of explorable uncertainty and is of intrinsic importance therein as it implies strong adversarial lower bounds for a wide range of interesting combinatorial problems such as knapsack and matchings. We consider a stochastic problem variant and give algorithms that, in expectation, improve upon these adversarial lower bounds. The key to our results is to prove a strong structural connection to a seemingly unrelated covering problem with uncertainty in the constraints via a linear programming formulation. We exploit this connection to derive an algorithmic framework that can be used to solve both problems under uncertainty, obtaining nearly tight bounds on the competitive ratio. This is the first non-trivial stochastic result concerning the sum of unknown values without further structure known for the set. With our novel methods, we lay the foundations for solving more general problems in the area of explorable uncertainty.

翻译：给定不确定值的若干子集，我们研究通过尽可能少的查询来识别总价值（不确定值之和）最小子集的问题。该集合选择问题属于可探索不确定性领域，并在此领域中具有本质重要性，因为它为背包、匹配等广泛有趣的组合问题提供了强对抗性下界。我们考虑一个随机问题变体，并给出在期望意义上优于这些对抗性下界的算法。我们结果的关键在于，通过线性规划表述，证明该问题与一个看似无关的、具有不确定性约束的覆盖问题之间存在强结构关联。我们利用这一关联推导出一个算法框架，该框架可用于解决这两个不确定性问题，并获得近乎紧的竞争比界。这是关于未知值总和（且已知集合无其他结构）的首个非平凡随机结果。我们通过新颖方法，为求解可探索不确定性领域中更一般的问题奠定了基础。