The Competition Complexity of Prophet Secretary

TL;DR

This paper analyzes the competition complexity of various online algorithms for the prophet secretary problem, revealing how different classes of algorithms perform in resource augmentation scenarios and establishing tight bounds for each class.

Contribution

It provides tight bounds on the competition complexity for single-threshold, time-based, activation-based, and general algorithms in the prophet secretary problem.

Findings

Single-threshold algorithms have a $ heta( ext{ln}(1/\epsilon))$ competition complexity.

Time-based and activation-based algorithms have a sub-optimal $ heta(\text{ln}(1/\epsilon)/\text{ln}\text{ln}(1/\epsilon))$ complexity.

General adaptive algorithms achieve a $ heta(\sqrt{\text{ln}(1/\epsilon)})$ competition complexity.

Abstract

We study the classic single-choice prophet secretary problem through a resource augmentation lens. Our goal is to bound the $(1-\epsilon)$-competition complexity for different classes of online algorithms. This metric asks for the smallest $k$ such that the expected value of the online algorithm on $k$ copies of the original instance, is at least a $(1 - \epsilon)$-approximation to the expected offline optimum on the original instance (without added copies). We consider four natural classes of online algorithms: single-threshold, time-based threshold, activation-based, and general algorithms. We show that for single-threshold algorithms the $(1-\epsilon)$-competition complexity is $\Theta(\ln(\frac{1}{\epsilon}))$ (as in the i.i.d. case). Additionally, we demonstrate that time-based threshold and activation-based algorithms (which cover all previous approaches for obtaining competitive-ratios for the classic prophet secretary problem) yield a sub-optimal $(1-\epsilon)$-competition complexity of $\Theta\left(\frac{\ln(\frac{1}{\epsilon})}{\ln\ln(\frac{1}{\epsilon})}\right)$, which is strictly better than the class of single-threshold algorithms. Finally, we find that the $(1-\epsilon)$-competition complexity of general adaptive algorithms is $\Theta(\sqrt{\ln(\frac{1}{\epsilon})})$, which is in sharp contrast to $\Theta(\ln\ln(\frac{1}{\epsilon}))$ in the i.i.d. case.