We consider search problems with nonobligatory inspection and single-item or combinatorial selection. A decision maker is presented with a number of items, each of which contains an unknown price, and can pay an inspection cost to observe the item's price before selecting it. Under single-item selection, the decision maker must select one item; under combinatorial selection, the decision maker must select a set of items that satisfies certain constraints. In our nonobligatory inspection setting, the decision maker can select items without first inspecting them. It is well-known that search with nonobligatory inspection is harder than the well-studied obligatory inspection case, for which the optimal policy for single-item selection (Weitzman, 1979) and approximation algorithms for combinatorial selection (Singla, 2018) are known. We introduce a technique, local hedging, for constructing policies with good approximation ratios in the nonobligatory inspection setting. Local hedging transforms policies for the obligatory inspection setting into policies for the nonobligatory inspection setting, at the cost of an extra factor in the approximation ratio. The factor is instance-dependent but is at most 4/3. We thus obtain the first approximation algorithms for a variety of combinatorial selection problems, including matroid basis, matching, and facility location.

翻译：本文研究具有非强制检查特性及单项或组合选择需求的搜索问题。决策者面对若干物品，每个物品包含未知价格，可通过支付检查成本在选定前观测其价格。在单项选择场景中，决策者必须选择一件物品；在组合选择场景中，决策者需选择满足特定约束的物品集合。在非强制检查设定下，决策者可在未检查物品的情况下直接进行选择。众所周知，非强制检查搜索问题比经过深入研究的强制检查情形更为复杂——后者在单项选择方面存在最优策略（Weitzman, 1979），在组合选择方面存在近似算法（Singla, 2018）。我们提出一种称为局部对冲的技术，用于构建在非强制检查设定下具有良好近似比的策略。该技术可将强制检查设定下的策略转化为非强制检查设定下的策略，其代价是引入额外的近似比系数。该系数虽依赖于具体问题实例，但始终不超过4/3。由此我们首次为多种组合选择问题（包括拟阵基、匹配和设施选址问题）提供了近似算法。