Revealing Optimal Thresholds for Generalized Secretary Problem via Continuous LP: Impacts on Online K-Item Auction and Bipartite K-Matching with Random Arrival Order

Abstract

We consider the general (J, K)-secretary problem, where n totally ordered items arrive in a random order. An algorithm observes the relative merits of arriving items and is allowed to make J selections. The objective is to maximize the expected number of items selected among the K best items.Buchbinder, Jain and Singh proposed a finite linear program (LP) that completely characterizes the problem, but it is difficult to analyze the asymptotic behavior of its optimal solution as n tends to infinity. Instead, we prove a formal connection between the finite model and an infinite model, where there are a countably infinite number of items, each of which has arrival time drawn independently and uniformly from [0, 1].The finite LP extends to a continuous LP, whose complementary slackness conditions reveal an optimal algorithm which involves JK thresholds that play a similar role as the 1 e -threshold in the optimal classical secretary algorithm. In particular, for the case K = 1, the J optimal thresholds have a nice "rational description". Our continuous LP analysis gives a very clear perspective on the problem, and the new insights inspire us to solve two related problems.1. We settle the open problem whether algorithms based only on relative merits can achieve optimal ratio for matroid secretary problems. We show that, for online 2-item auction with random arriving bids (the K-uniform matroid problem with K = 2), an algorithm making decisions based only on relative merits cannot achieve the optimal ratio. This is in contrast with the folklore that, for online 1-item auction, no algorithm can have performance ratio strictly larger than 1 e , which is achievable by an algorithm that considers only relative merits. 2. We give a general transformation technique that takes any monotone algorithm (such as threshold

The Secretary Problem with Independent Sampling

“…The approach of approximating the problem when n is large by a continuous time problem was pioneered by Bruss [5] and has been used for different optimal stopping problems (see e.g. [7,16]). Lemma 3.1.…”

The solution of a generalized secretary problem via analytic expressions

“…For example, in [3] the authors consider a so-called J-choice K-best secretary problem (the case J = 1 was the subject of the present paper), where finding of an optimal solution reduces to solving the corresponding linear program. In [4] the authors use linear programming but to the so-called continuous and infinite models of the secretary problem (see also [2,12]). In [15] even a more general problem is studied via this technique -a so called shared Q-queue J-choice K-best secretary problem.…”

“…The proof of the next proposition is based on various properties of the maps D l , c l (l ∈ [k] 0 ), which we derive in Section 6 (see Lemmas 3,4).…”

Strong Algorithms for the Ordinal Matroid Secretary Problem

“…In the J-choice K-best secretary problem by Buchbinder et al [10], one has to select J elements from a randomly ordered stream getting profit only from the top K elements. In particular, every α competitive algorithm for the ρ-choice ρ-best secretary problem that only uses ordinal information, is an α intersection-competitive algorithm for uniform matroids of rank ρ. Chan et al [12] show that the best competitive ratio achievable by an algorithm that can only use ordinal information on the 2-choice 2-best secretary problem is approximately 0.488628. In contrast, by using numerical information one can achieve a higher competitive ratio of 0.492006 that also works for the sumof-weights objective function (i.e., for utility-competitiveness).…”

“…This assumption can be removed by using standard arguments; we can set ρ equal to twice the rank of the sampled part 11. The original analysis uses half of n, but the analysis gets simpler if one uses Bin(n, 1/2) since one can assume that each element is in the sample with probability 1/2 independent of the rest 12. A loop in a matroid is an element that belongs to no basis.…”