Secretary problems: Weights and discounts

The classical secretary problem studies the problem of selecting online an element (a "secretary") with maximum value in a randomly ordered sequence. The difficulty lies in the fact that an element must be either selected or discarded upon its arrival, and this decision is irrevocable. Constant-competitive algorithms are known for the classical secretary problems (see, e.g., the survey of Freeman [7]) and several variants. We study the following two extensions of the secretary problem: • In the discounted secretary problem, there is a time-dependent "discount" factor d(t). and the benefit derived from selecting an element/secretary e at time t is d(t)·v(e). For this problem with arbitrary (not necessarily decreasing) functions d(t), we show a constant-competitive algorithm when the expected optimum is known in advance. With no prior knowledge, we exhibit a lower bound of Ω(log n/log log n), and give a nearly-matching O(logn)-competitive algorithm. • In the weighted secretary problem, up to K secretaries can be selected; when a secretary is selected (s)he must be irrevocably assigned to one of K positions, with position k having weight w(k), and assigning object/secretary e to position k has benefit w(k)·v(e). The goal is to select secretaries and assign them to positions to maximize ∑_e,k w(k)·v(e)·x_ek where x_ek is an indicator variable that secretary e is assigned position k. We give constant-competitive algorithms for this problem. Most of these results can also be extended to the matroid secretary case (Babaioff et al. [2]) for a large family of matroids with a constant-factor loss, and an O(log rank) loss for general matroids. These results are based on a reduction from various matroids to partition matroids which present a unified approach to many of the upper bounds of Babaioff et al. These problems have connections to online mechanism design (see, e.g., Hajiaghayi et al. [9]). All our algorithms are monotone, and hence lead to truthful mechanisms for the corresponding online auction problems.