Online Pen Testing

TL;DR

This paper introduces algorithms for a pen testing problem where the goal is to select the pen with the most ink using limited information, achieving near-optimal results in both prophet and secretary models.

Contribution

The paper develops robust algorithms for the pen testing problem that work with minimal distributional information and provide optimal competitive ratios in prophet and secretary settings.

Findings

Achieves $O(\log n)$ approximation for pen testing.

Designs algorithms requiring only one sample per distribution.

Provides optimal competitive ratios in both models.

Abstract

We study a "pen testing" problem, in which we are given $n$ pens with unknown amounts of ink $X_1, X_2, \ldots, X_n$, and we want to choose a pen with the maximum amount of remaining ink in it. The challenge is that we cannot access each $X_i$ directly; we only get to write with the $i$-th pen until either a certain amount of ink is used, or the pen runs out of ink. In both cases, this testing reduces the remaining ink in the pen and thus the utility of selecting it. Despite this significant lack of information, we show that it is possible to approximately maximize our utility up to an $O(\log n)$ factor. Formally, we consider two different setups: the "prophet" setting, in which each $X_i$ is independently drawn from some distribution $\mathcal{D}_i$, and the "secretary" setting, in which $(X_i)_{i=1}^n$ is a random permutation of arbitrary $a_1, a_2, \ldots, a_n$. We derive the optimal competitive ratios in both settings up to constant factors. Our algorithms are surprisingly robust: (1) In the prophet setting, we only require one sample from each $\mathcal{D}_i$, rather than a full description of the distribution; (2) In the secretary setting, the algorithm also succeeds under an arbitrary permutation, if an estimate of the maximum $a_i$ is given. Our techniques include a non-trivial online sampling scheme from a sequence with an unknown length, as well as the construction of a hard, non-uniform distribution over permutations. Both might be of independent interest. We also highlight some immediate open problems and discuss several directions for future research.