Who Solved the Secretary Problem?

Ferguson, Thomas S.

doi:10.1214/ss/1177012493

Cited by 582 publications

(438 citation statements)

References 42 publications

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“…The objective is to maximize the probability that this value is the maximum value, or, equivalently, the expectation of the ratio of this value and the ex post maximum value in the sequence. As observed by Ferguson [3], this formulation of the problem is equivalent to the original one, and the same aspiration strategy achieves a factor of 1/e. Furthermore, this result is tight, that is, there is a distribution of values for which, even knowing the distribution, the optimal strategy is the aspiration strategy with a factor of 1/e.…”

Section: Introductionmentioning

confidence: 68%

“…An alternative formulation of the secretary problem (and according to Ferguson [3], the formulation closer to the original "game of googol" puzzle due to Martin Gardner in 1960 [5]) is as follows: An adversary selects the distribution from which the n values will be drawn independently. 1 The algorithm learns about this distribution (but not the actual values), and then sees the values one by one.…”

Section: Introductionmentioning

confidence: 99%

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The Secretary Problem with a Hazard Rate Condition

Mahdian

McAfee

Pennock

2008

Lecture Notes in Computer Science

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Abstract. In the classical secretary problem, the objective is to select the candidate of maximum value among a set of n candidates arriving one by one. The value of the candidates come from an unknown distribution and is revealed at the time the candidate arrives, at which point an irrevocable decision on whether to select the candidate must be made. The well-known solution to this problem, due to Dynkin, waits for n/e steps to set an "aspiration level" equal to the maximum value of the candidates seen, and then accepts the first candidate whose value exceeds this level. This guarantees a probability of at least 1/e of selecting the maximum value candidate, and there are distributions for which this is essentially the best possible. One feature of this algorithm that seems at odds with reality is that it prescribes a long waiting period before selecting a candidate. In this paper, we show that if a standard hazard rate condition is imposed on the distribution of values, the waiting period falls from n/e to n/ log(n), meaning that it is enough to observe a diminishingly small sample to set the optimal aspiration level. This result is tight, as both the hazard condition and the optimal sampling period bind exactly for the exponential distribution.

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Section: Introductionmentioning

confidence: 68%

Section: Introductionmentioning

confidence: 99%

The Secretary Problem with a Hazard Rate Condition

Mahdian

McAfee

Pennock

2008

Lecture Notes in Computer Science

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“…The secretary problem is a classical problem in probability theory, with obscure origins in the 1950's and early 60's ( [11,17,8]; see also [10]). Here, the goal is to select the best candidate out of a sequence revealed one-by-one, where the ranking is uniformly random.…”

Section: Introductionmentioning

confidence: 99%

“…Here, the goal is to select the best candidate out of a sequence revealed one-by-one, where the ranking is uniformly random. A classical solution finds the best candidate with probability at least 1/e [10]. Over the years a number of variants have been studied, starting with [12] where multiple choices and various measures of success were considered for the first time.…”

Section: Introductionmentioning

confidence: 99%

On Variants of the Matroid Secretary Problem

Gharan

Vondrák

2011

Algorithms – ESA 2011

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Abstract. We present a number of positive and negative results for variants of the matroid secretary problem. Most notably, we design a constant-factor competitive algorithm for the "random assignment" model where the weights are assigned randomly to the elements of a matroid, and then the elements arrive on-line in an adversarial order (extending a result of Soto [20]). This is under the assumption that the matroid is known in advance. If the matroid is unknown in advance, we present an O(log r log n)-approximation, and prove that a better than O(log n/ log log n) approximation is impossible. This resolves an open question posed by Babaioff et al. [3]. As a natural special case, we also consider the classical secretary problem where the number of candidates n is unknown in advance. If n is chosen by an adversary from {1, . . . , N }, we provide a nearly tight answer, by providing an algorithm that chooses the best candidate with probability at least 1/(HN−1 + 1) and prove that a probability better than 1/HN cannot be achieved (where HN is the N -th harmonic number).

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“…Perhaps the most well studied matroid secretary problem is the secretary problem, which first appeared as a folklore problem in the 1950's and has a long history [4,5]. The problem was first solved by Lindley, who also presents a competitive algorithm for the secretary problem [11].…”

Section: Introductionmentioning

confidence: 99%

Competitive Weighted Matching in Transversal Matroids

Dimitrov

Plaxton

2008

Automata, Languages and Programming

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Consider a bipartite graph with a set of left-vertices and a set of right-vertices. All the edges adjacent to the same left-vertex have the same weight. We present an algorithm that, given the set of right-vertices and the number of left-vertices, processes a uniformly random permutation of the left-vertices, one left-vertex at a time. In processing a particular left-vertex, the algorithm either permanently matches the left-vertex to a thus-far unmatched right-vertex, or decides never to match the left-vertex. The weight of the matching returned by our algorithm is within a constant factor of that of a maximum weight matching, generalizing the recent results of Babaioff et al. * Supported by an MCD Fellowship from the University of Texas at Austin.

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Who Solved the Secretary Problem?

Cited by 582 publications

References 42 publications

The Secretary Problem with a Hazard Rate Condition

The Secretary Problem with a Hazard Rate Condition

On Variants of the Matroid Secretary Problem

Competitive Weighted Matching in Transversal Matroids

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