Competitive Analysis with a Sample and the Secretary Problem

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 competitiveratio 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.

Secretaries with Advice

“…We unify disparate information augmentation settings for the secretary problem into a single framework that is rich enough to capture the classic random arrival model [13], the Gilbert-Mosteller i.i.d. setting [16], its Markovian generalizations [2,12,18,34], and the recently introduced sample-based variants [9,21].…”

“…We highlight some of the results that can be derived using our framework. First we focus on secretary algorithms for the sampling model of Kaplan et al [21]. In this model an adversary chooses + numbers, a random subset of of these numbers are presented to the algorithm as samples at the outset, the remaining numbers are presented to the algorithm oneby-one, in random order.…”

“…Another interesting line of work in this area studies the secretary and prophet problems in the presence of a limited number of samples [9-11, 21, 32]. Most relevant in this context are the Kaplan et al paper [21], which is the model we adopt here, and the paper by Correa et al [9] as it is the only prior work that looks at the secretary objective.…”

“…Subsequent work has identified this as the indifference case of what has become to be known as the "bang bang principle" [2,12,34]: In a random walk that is started at zero and in which the probability to move up by one is and the probability to move down by one is 1 − it is best to stop immediately when < 1/2 and to wait until the end when > 1/2. Example 4 (Secretaries with Samples [9,21]). An adversary writes down + numbers.…”

“…Indeed there has been a flurry of recent work on stopping problems, especially the prophet inequality variety, with limited information about an underlying distribution in the form of samples (e.g., [4, 9-11, 21, 32]). In a model popularized by Kaplan et al [21], for example, an adversary chooses + numbers. A random subset of of these numbers are revealed to the decision maker (as samples) at the outset.…”