随机探测与选择停止问题的高效近似方案
Efficient Approximation Schemes for Stochastic Probing and Selection-Stopping Problems
AI总结：
本文提出通用框架，通过向多维 Santa Claus 问题归约，为随机探测与选择停止问题设计高效多项式时间近似方案（EPTAS），改进了现有结果。
摘要：
在本文中，我们提出了一个通用框架，为基础的随机组合优化问题设计高效多项式时间近似方案（EPTAS）。给定误差参数 $ε>0$，此类算法方案在 $t(ε)\cdot poly(|{\cal I}|)$ 时间内实现 $(1-ε)$-近似，其中 $t(\cdot)$ 是仅依赖于 $ε$ 的函数，$|{\cal I}|$ 表示输入长度。从技术上讲，我们的方法依赖于提出针对新引入的多维 Santa Claus 问题的定制化归约。尽管该问题的单维版本已知是 APX-Hard 的，但我们证明，对于常数数量的机器和维度，可以设计 EPTAS，这适用于我们的每个应用。为展示框架的通用性，我们首先研究选择停止场景，为 Free-Order Prophets 问题 [Agrawal et al., EC~'20] 及其成本驱动的一般化 Pandora's Box with Commitment [Fu et al., ICALP~'18] 导出 EPTAS。这些结果构成了非自适应设置中的首个近似方案，并改进了其自适应变体已知的低效多项式时间近似方案（PTAS）。接下来，将注意力转向随机探测问题，我们获得了自适应 ProbeMax 问题及其非自适应对应问题的 EPTAS；在这两种情况下，最先进的可近似性结果一直是低效的 PTASes [Chen et al., NIPS~'16; Fu et al., ICALP~'18]。
英文摘要：
In this paper, we propose a general framework to design {efficient} polynomial time approximation schemes (EPTAS) for fundamental stochastic combinatorial optimization problems. Given an error parameter $ε>0$, such algorithmic schemes attain a $(1-ε)$-approximation in $t(ε)\cdot poly(|{\cal I}|)$ time, where $t(\cdot)$ is a function that depends only on $ε$ and $|{\cal I}|$ denotes the input length. Technically speaking, our approach relies on presenting tailor-made reductions to a newly-introduced multi-dimensional Santa Claus problem. Even though the single-dimensional version of this problem is already known to be APX-Hard, we prove that an EPTAS can be designed for a constant number of machines and dimensions, which hold for each of our applications. To demonstrate the versatility of our framework, we first study selection-stopping settings to derive an EPTAS for the Free-Order Prophets problem [Agrawal et al., EC~'20] and for its cost-driven generalization, Pandora's Box with Commitment [Fu et al., ICALP~'18]. These results constitute the first approximation schemes in the non-adaptive setting and improve on known \emph{inefficient} polynomial time approximation schemes (PTAS) for their adaptive variants. Next, turning our attention to stochastic probing problems, we obtain an EPTAS for the adaptive ProbeMax problem as well as for its non-adaptive counterpart; in both cases, state-of-the-art approximability results have been inefficient PTASes [Chen et al., NIPS~'16; Fu et al., ICALP~'18].
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