Secretary Problems with Non-Uniform Arrival Order

Abstract

For a number of problems in the theory of online algorithms, it is known that the assumption that elements arrive in uniformly random order enables the design of algorithms with much better performance guarantees than under worst-case assumptions. The quintessential example of this phenomenon is the secretary problem, in which an algorithm attempts to stop a sequence at the moment it observes the maximum value in the sequence. As is well known, if the sequence is presented in uniformly random order there is an algorithm that succeeds with probability 1/e, whereas no non-trivial performance guarantee is possible if the elements arrive in worst-case order.In many of the applications of online algorithms, it is reasonable to assume there is some randomness in the input sequence, but unreasonable to assume that the arrival ordering is uniformly random. This work initiates an investigation into relaxations of the random-ordering hypothesis in online algorithms, by focusing on the secretary problem and asking what performance guarantees one can prove under relaxed assumptions. Toward this end, we present two sets of properties of distributions over permutations as sufficient conditions, called the (p, q, δ)-block-independence property and (k, δ)-uniform-induced-ordering property. We show these two are asymptotically equivalent by borrowing some techniques from the celebrated approximation theory. Moreover, we show they both imply the existence of secretary algorithms with constant probability of correct selection, approaching the optimal constant 1/e as the related parameters of the property tend towards their extreme values. Both of these properties are significantly weaker than the usual assumption of uniform randomness; we substantiate this by providing several constructions of distributions that satisfy (p, q, δ)-block-independence. As one application of our investigation, we prove that Θ(log log n) is the minimum entropy of any permutation distribution that permits constant probability of correct selection in the secretary problem with n elements. While our block-independence condition is sufficient for constant probability of correct selection, it is not necessary; however, we present complexity-theoretic evidence that no simple necessary and sufficient criterion exists. Finally, we explore the extent to which the performance guarantees of other algorithms are preserved when one relaxes the uniform random ordering assumption to (p, q, δ)-block-independence, obtaining a positive result for Kleinberg's multiple-choice secretary algorithm and a negative result for the weighted bipartite matching algorithm of Korula

Beyond matroids: secretary problem and prophet inequality with general constraints

“…Even the 2013 survey by Dinitz [4] seems outdated with so much exciting progress in the last couple of years (e.g. [1,8,11,17]).…”

Combinatorial Prophet Inequalities

“…Obtaining a constant competitive ratio for general matroids remains a central open problem, but constant bounds are known for many special cases (see survey by Dinitz [19] and references therein). Other variants have also been considered, such as a hiring that returns for a second (or potentially k-th) interview [50], or secretaries whose order of arrival has exceptionally low entropy [37]. Of special interest to us is a recent paper by Feldman and Izsak [25] Prophet Inequality The connection between multiple-choice prophet inequalities and mechanism design was recognized in the seminal paper by Hajiaghayi et al [33].…”

Strong Algorithms for the Ordinal Matroid Secretary Problem

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