🤖 AI Summary
This work addresses five classic stochastic combinatorial optimization problems—Free-Order Prophet Inequalities, Pandora’s Box with Commitment, and Adaptive/Non-adaptive ProbeMax—and presents the first unified, efficient polynomial-time approximation scheme (EPTAS): for any ε > 0, it achieves a (1−ε)-approximation in t(ε)·poly(|I|) time. Departing from prior approaches constrained by single-dimensional APX-hardness, the authors introduce the multi-dimensional Santa Claus problem as a novel reduction target. Their method features problem-specific reductions, multi-dimensional resource modeling, structured dynamic programming, and error-controlled discretization. The result yields the first practical EPTAS for all five problems, with the ε-dependent factor fully decoupled from input size—marking a substantial improvement over previous existential, non-scalable, and computationally inefficient PTASes. Notably, this is the first efficient approximation scheme for the non-adaptive setting, closing a long-standing gap in the literature.
📝 Abstract
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 $epsilon>0$, such algorithmic schemes attain a $(1-epsilon)$-approximation in $t(epsilon)cdot poly(|{cal I}|)$ time, where $t(cdot)$ is a function that depends only on $epsilon$ 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].
