# Competitive Analysis with a Sample and the Secretary Problem

## TL;DR

This paper introduces a new online model that incorporates prior sample information, bridging worst-case and stochastic models, and applies it to the secretary problem to develop optimal algorithms for different scenarios.

## Contribution

It extends online models by revealing a sample beforehand and provides optimal algorithms for the secretary problem in these new settings.

## Key findings

- Optimal competitive ratios achieved in worst-case model with any sample size.
- Almost tight competitive ratios for small samples in the random-order model.
- No single algorithm is optimal in both models for large samples.

## Abstract

We extend the standard online worst-case model to accommodate past experience which is available to the online player in many practical scenarios. We do this by revealing a random sample of the adversarial input to the online player ahead of time. The online player competes with the expected optimal value on the part of the input that arrives online. Our model bridges between existing online stochastic models (e.g., items are drawn i.i.d. from a distribution) and the online worst-case model. We also extend in a similar manner (by revealing a sample) the online random-order model. We study the classical secretary problem in our new models. In the worst-case model we present a simple online algorithm with optimal competitive-ratio for any sample size. In the random-order model, we also give a simple online algorithm with an almost tight competitive-ratio for small sample sizes. Interestingly, we prove that for a large enough sample, no algorithm can be simultaneously optimal both in the worst-cast and random-order models.