Our main contribution is a general framework to design \emph{efficient} polynomial time approximation schemes (EPTAS) for fundamental stochastic combinatorial optimization problems. Given an error parameter $\epsilon>0$, such algorithmic schemes attain a $(1-\epsilon)$-approximation in $t(\epsilon)\cdot poly(n)$ time, where $t(\cdot)$ is some function that depends only on $\epsilon$. Technically speaking, our approach relies on presenting tailor-made reductions to a newly-introduced multi-dimensional load balancing problem. Even though the single-dimensional problem is already known to be APX-Hard, we prove that an EPTAS can be designed under certain structural assumptions, 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 {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].

翻译：我们的主要贡献是建立了一个通用框架，用于设计基础随机组合优化问题的多项式时间高效近似方案（EPTAS）。给定误差参数 $\epsilon>0$，此类算法方案能在 $t(\epsilon)\cdot poly(n)$ 时间内实现 $(1-\epsilon)$ 近似，其中 $t(\cdot)$ 是仅依赖于 $\epsilon$ 的函数。从技术角度看，我们的方法依赖于对新型多维负载均衡问题的定制化归约。尽管单维问题已被证明是 APX 困难的，但我们证明在特定结构假设下可设计 EPTAS，这些假设在我们的所有应用场景中均成立。为展示框架的通用性，我们首先研究选择停止设定，为自由顺序先知问题 [Agrawal 等，EC'20] 及其代价驱动推广——带承诺的潘多拉魔盒 [Fu 等，ICALP'18] 推导 EPTAS。这些结果构成了非自适应设定下的首个近似方案，并改进了其自适应变体中已知的多项式时间近似方案（PTAS）。随后针对随机探测问题，我们为自适应 ProbeMax 问题及其非自适应变体均获得 EPTAS；在两种情形中，现有最优近似结果均为非高效 PTAS [Chen 等，NIPS'16; Fu 等，ICALP'18]。