Efficient Approximation Schemes for Stochastic Probing and Prophet Problems.

Our main contribution is a general framework to design efficient polynomial time approximation schemes (EPTAS) for fundamental classes of stochastic combinatorial optimization problems. Given an error parameter $\epsilon>0$, such algorithmic schemes attain a $(1+\epsilon)$-approximation in only $t(\epsilon)\cdot poly(n)$ 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 extension of the Santa Claus problem [Bansal-Sviridenko, STOC'06]. 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 our applications. To demonstrate the versatility of our framework, 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 polynomial time approximation schemes (PTAS) [Chen et al., NIPS'16; Fu et al., ICALP'18]. Turning our attention to selection-stopping settings, we further 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 improve on known PTASes for their adaptive variants, and constitute the first non-trivial approximations in the non-adaptive setting.