随机多值探测的（近）最优适应间隙
(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing
摘要：
考虑一个肾脏交换应用程序，我们希望在随机图中找到最大匹配。为了发现边缘$e$是否存在，我们需要执行昂贵的测试，在这种情况下，边缘$e$以\emph{known}概率$p_e$独立出现。给定测试的总成本预算，我们的目标是找到一个测试策略，最大化预期的最大匹配大小。上述应用是随机探测问题的一个例子。一般来说，最佳的随机探测策略是很难找到的，因为它是\emph{adaptive} -根据探测边的结果决定下一个要探测的边。另一种方法是显示\emph{adaptivity gap}很小，即，最佳\emph{non-adaptive}策略总是具有接近于最佳自适应策略的值。这使我们能够专注于设计简单得多的非适应性策略。然而，以前的作品都集中在伯努利随机变量，只能捕捉边缘是否出现。在这项工作中，我们引入了一个多值随机探测问题，它也可以模拟的情况下，边的权重有一个概率分布在多个值。我们的主要技术贡献是获得（最坏情况下）的适应性差距（近）的最佳界限，多值随机探测前缀封闭的约束。对于单调子模函数，我们证明了自适应间隙至多为$2$，并给出了一个匹配下界。对于$k$ -可扩系统（$k$拟阵的交的推广）的加权秩函数，我们证明了自适应间隙在$O(k\log k)$和$k$之间.这些结果都是未知的，即使在伯努利的情况下，我们的上限和下限也适用，从而解决了一个悬而未决的问题古普塔等人。
英文摘要：
Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge $e$ exists, we need to perform an expensive test, in which case the edge $e$ appears independently with a \emph{known} probability $p_e$. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size. The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is \emph{adaptive}---decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the \emph{adaptivity gap} is small, i.e., the best \emph{non-adaptive} strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values. Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most $2$ and provide a matching lower bound. For a weighted rank function of a $k$-extendible system (a generalization of intersection of $k$ matroids), we show the adaptivity gap is between $O(k\log k)$ and $k$. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
