No 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

Hierarchical Clustering (HC) is a widely studied problem in exploratory data analysis, usually tackled by simple agglomerative procedures like average-linkage, single-linkage or complete-linkage. In this paper we focus on two objectives, introduced recently to give insight into the performance of average-linkage clustering: a similarity based HC objective proposed by Moseley and Wang (2017) and a dissimilarity based HC objective proposed by . In both cases, we present tight counterexamples showing that average-linkage cannot obtain better than 1 3 and 2 3 approximations respectively (in the worst-case), settling an open question raised in Moseley and Wang (2017). This matches the approximation ratio of a random solution, raising a natural question: can we beat average-linkage for these objectives? We answer this in the affirmative, giving two new algorithms based on semidefinite programming with provably better guarantees.

We provide a polynomial time reduction from Bayesian incentive compatible mechanism design to Bayesian algorithm design for welfare maximization problems. Unlike prior results, our reduction achieves exact incentive compatibility for problems with multi-dimensional and continuous type spaces. The key technical barrier preventing exact incentive compatibility in prior black-box reductions is that repairing violations of incentive constraints requires understanding the distribution of the mechanism’s output, which is typically #P-hard to compute. Reductions that instead estimate the output distribution by sampling inevitably suffer from sampling error, which typically precludes exact incentive compatibility. We overcome this barrier by employing and generalizing the computational model in the literature on Bernoulli Factories . In a Bernoulli factory problem, one is given a function mapping the bias of an “input coin” to that of an “output coin,” and the challenge is to efficiently simulate the output coin given only sample access to the input coin. This is the key ingredient in designing an incentive compatible mechanism for bipartite matching, which can be used to make the approximately incentive compatible reduction of Hartline et al. [18] exactly incentive compatible.

For selling a single item to agents with independent but non-identically distributed values, the revenue optimal auction is complex. With respect to it, Hartline and Roughgarden (2009) showed that the approximation factor of the second-price auction with an anonymous reserve is between two and four. We consider the more demanding problem of approximating the revenue of the ex ante relaxation of the auction problem by posting an anonymous price (while supplies last) and prove that their worst-case ratio is e. As a corollary, the upper-bound of anonymous pricing or anonymous reserves versus the optimal auction improves from four to e. We conclude that, up to an e factor, discrimination and simultaneity are unimportant for driving revenue in single-item auctions.