Efficient Approximation Schemes for Stochastic Probing and Selection-Stopping Problems

TL;DR

This paper introduces a general framework for designing efficient polynomial time approximation schemes for stochastic combinatorial optimization problems, leveraging reductions to a multi-dimensional Santa Claus problem, and applies it to various selection-stopping and probing problems.

Contribution

The paper presents the first EPTAS for several stochastic optimization problems, improving upon previous inefficient PTAS and introducing a novel reduction approach to a multi-dimensional Santa Claus problem.

Findings

EPTAS for selection-stopping problems like Free-Order Prophets and Pandora's Box with Commitment.

EPTAS for stochastic probing problems such as ProbeMax, both adaptive and non-adaptive.

Reduction to multi-dimensional Santa Claus problem enables efficient approximation schemes.

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].