Efficient Approximation Schemes for Stochastic Probing and Selection-Stopping Problems

TL;DR

This work develops an overarching framework to design efficient EPTAS for a broad class of stochastic probing and selection-stopping problems by reducing them to a Multi-Dimensional Santa Claus problem with a constant number of machines and dimensions. The authors deliver an array of results: an EPTAS for Free-Order Prophets and its Pandora's Box with Commitment generalization (non-adaptive), and EPTASes for both adaptive and non-adaptive ProbeMax, improving on prior inefficient PTASs. The core technique combines bucketing of the optimal solution into a small number of types with additive hyper-parameters, a careful guessing step, a strengthened LP, and a dependent-rounding scheme that preserves cardinality while controlling constraint violations. These reductions yield near-optimal policies and selections with provable guarantees, and they extend to equivalences between Pandora's Box with Commitment and Free-Order Prophets as well as multi-element selection. The results have significant practical impact for stochastic optimization under uncertainty, offering instance-independent, scalable approximation schemes for complex adaptive and non-adaptive settings.

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 $ε>0$, 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].

Figures (1)

• Figure 1: An illustration of our bucketing scheme.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Informal Theorem \ref{['thm:result_santa_claus']}
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Claim 2.4
• Claim 2.5