Abstract
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2009 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Secretary Problems: Weights and DiscountsMoshe Babaioff, Michael Dinitz, Anupam Gupta, Nicole Immorlica, and Kunal TalwarMoshe Babaioff, Michael Dinitz, Anupam Gupta, Nicole Immorlica, and Kunal Talwarpp.1245 - 1254Chapter DOI:https://doi.org/10.1137/1.9781611973068.135PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract 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 , and give a nearly-matching O (log n)-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 Σe,k w(k) · v(e) · xek where xek 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. Previous chapter Next chapter RelatedDetails Published:2009ISBN:978-0-89871-680-1eISBN:978-1-61197-306-8 https://doi.org/10.1137/1.9781611973068Book Series Name:ProceedingsBook Code:PR132Book Pages:xviii + 1288
Highlights
The classical secretary problem [5, 7] captures the question of finding the element with the maximum value in an online fashion, when the elements are presented in a random order.It is well known that waiting until one sees 1/e fraction of the elements, and picking the first element attaining a value greater than the maximum value seen in the first 1/e fraction of the elements gives an e-competitive algorithm, and this is the best possible
“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 functions d(t), we show a constant-competitive algorithm when the expected optimum is known in advance
The problem is of interest due to its connections with online mechanism design: if we have a single good to sell and agents with varying valuations for that object arriving online, the secretary problem captures the difficulty in picking the person with the largest valuation for that good [9, 10]
Summary
The classical secretary problem [5, 7] captures the question of finding the element with the maximum value in an online fashion, when the elements are presented in a random order. The elements of the secretary problem are agents and the element value is the agent’s value for the good; the goal of the mechanism designer is to maximize social welfare, or sell the good to the agent with the highest valuation Another application is to modeling the economic decision facing an agent who wishes to select one of an online sequence of goods—e.g., an agent buying a house or a company hiring an employee. When seeking to purchase a house, we might think of choosing a slightly suboptimal house at the beginning of the experiment (and being able to occupy it for the entire period) as being more desirable than a long wait to pick the most desirable house We model such a problem as the discounted secretary problem, where we are given “discount” values d(t) for every time step t: the benefit derived from choosing an element with value v(e) at time t is the product d(t) · v(e). If the elements arrive in a random order, and assigning good k to agent e accrues a benefit of v(e) · w(k), how should we choose K agents and assign goods to them to maximize the total expected benefit?
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