We consider max-weighted matching with costs for learning the weights, modeled as a "Pandora's Box" on each endpoint of an edge. Each vertex has an initially-unknown value for being matched to a neighbor, and an algorithm must pay some cost to observe this value. The goal is to maximize the total matched value minus costs. Our model is inspired by two-sided settings, such as matching employees to employers. Importantly for such settings, we allow for negative values which cause existing approaches to fail. We first prove upper bounds for algorithms in two natural classes. Any algorithm that "bundles" the two Pandora boxes incident to an edge is an $o(1)$-approximation. Likewise, any "vertex-based" algorithm, which uses properties of the separate Pandora's boxes but does not consider the interaction of their value distributions, is an $o(1)$-approximation. Instead, we utilize Pandora's Nested-Box Problem, i.e. multiple stages of inspection. We give a self-contained, fully constructive optimal solution to the nested-boxes problem, which may have structural observations of interest compared to prior work. By interpreting each edge as a nested box, we leverage this solution to obtain a constant-factor approximation algorithm. Finally, we show any "edge-based" algorithm, which considers the interactions of values along an edge but not with the rest of the graph, is also an $o(1)$-approximation.

翻译：我们考虑在需要学习权重的情况下进行最大权重匹配，其中每个边的端点被建模为一个“潘多拉盒”。每个顶点与邻居匹配的价值初始未知，算法需支付一定成本才能观测该价值。目标是在减去成本后最大化总匹配价值。我们的模型受到双边匹配场景（如员工与雇主的匹配）的启发。对此类场景至关重要的是，我们允许负价值的存在，而这会导致现有方法失效。我们首先证明了两类自然算法性能的上界。任何将边关联的两个潘多拉盒“捆绑”处理的算法，其近似比均为 $o(1)$。同样，任何“基于顶点”的算法（即仅利用各潘多拉盒的独立属性而未考虑其价值分布间的交互作用），其近似比也为 $o(1)$。为此，我们采用潘多拉嵌套盒问题（即多阶段探查模型）。我们给出了一种独立完备、完全构造性的嵌套盒问题最优解法，与先前工作相比，该方法可能具有值得关注的结构性洞见。通过将每条边解释为一个嵌套盒，我们利用该解法得到了一种常数因子近似算法。最后，我们证明任何“基于边”的算法（即考虑边上价值交互但忽略其与图中其余部分相互作用）的近似比同样为 $o(1)$。