The stochastic matching problem deals with finding a maximum matching in a graph whose edges are unknown but can be accessed via queries. This is a special case of stochastic k-set packing, where the problem is to find a maximum packing of sets, each of which exists with some probability. In this paper, we provide edge and set query algorithms for these two problems, respectively, that provably achieve some fraction of the omniscient optimal solution.Our main theoretical result for the stochastic matching (i.e., 2-set packing) problem is the design of an adaptive algorithm that queries only a constant number of edges per vertex and achieves a (1 − ) fraction of the omniscient optimal solution, for an arbitrarily small > 0. Moreover, this adaptive algorithm performs the queries in only a constant number of rounds. We complement this result with a non-adaptive (i.e., one round of queries) algorithm that achieves a (0.5 − ) fraction of the omniscient optimum. We also extend both our results to stochastic k-set packing by designing an adaptive algorithm that achieves a ( 2 k − ) fraction of the omniscient optimal solution, again with only O(1) queries per element. This guarantee is close to the best known polynomial-time approximation ratio of 3 k+1 − for the deterministic k-set packing problem [22].We empirically explore the application of (adaptations of) these algorithms to the kidney exchange problem, where patients with end-stage renal failure swap willing but incompatible donors. We show on both generated data and on real data from the first 169 match runs of the UNOS nationwide kidney exchange that even a very small number of non-adaptive edge queries per vertex results in large gains in expected successful matches. arXiv:1407.4094v2 [cs.DS] 29 Apr 2015 interest, then, are algorithms that first query some subset of edges to find the ones that exist, and based on these queries, produce a matching that is as large as possible. The stochastic matching problem is a special case of stochastic k-set packing, where each set exists only with some probability, and the problem is to find a packing of maximum size of those sets that do exist.Without any constraints, one can simply query all edges or sets, and then output the maximum matching or packing over those that exist-but this level of freedom may not always be available. We are interested in the tradeoff between the number of queries and the fraction of the omnsicient optimal solution achieved. Specifically, we ask: In order to perform as well as the omniscient optimum in the stochastic matching problem, do we need to query (almost) all the edges, that is, do we need a budget of Θ(n) queries per vertex, where n is the number of vertices? Or, can we, for any arbitrarily small > 0, achieve a (1 − ) fraction of the omniscient optimum by using an o(n) per-vertex budget? We answer these questions, as well as their extensions to the k-set packing problem. We support our theoretical results empirically on both generated and real data from a large fielded kidney exchange ...

We consider the design of computationally efficient online learning algorithms in an adversarial setting in which the learner has access to an offline optimization oracle. We present an algorithm called Generalized Follow-the-Perturbed-Leader and provide conditions under which it is oracle-efficient while achieving vanishing regret. Our algorithm generalizes the Follow-the-Perturbed-Leader (FTPL) approach of Kalai and Vempala [32] in which the action with the highest randomly perturbed historical performance is chosen on each round. FTPL is inefficient when the number of actions is exponential in the parameters of interest. Our algorithm creates more compact perturbations by augmenting the observed history with randomly generated synthetic history, and choosing the action with the (near) best performance on this augmented history. As we show, when certain structural properties hold, the augmented history is of polynomial size even when the learner's action space is exponential, yielding oracle-efficient learning. Our results make significant progress on an open problem raised by Hazan and Koren [27], who showed that oracle-efficient algorithms do not exist in general [26] and asked whether one can identify properties under which oracle-efficient online learning may be possible.Our second main contribution is the introduction of a new adversarial online auction-design framework for revenue maximization and the application of our oracle-efficient learning results to the adaptive design of auctions. In our framework, a seller repeatedly sells an item or set of items to a population of buyers by adaptively selecting auctions from a fixed target class. The goal of the seller is to leverage historical bid data to pick an auction on each iteration in such a way that the seller's overall revenue compares favorably with the revenue he would have obtained using the best auction from the class in hindsight. Since this is a specific case of adversarial online learning, we can apply our framework and provide new oracle-efficient learning results for: (1) Vickrey-Clarkes-Groves (VCG) auctions with bidder-specific reserves in single-parameter settings, (2) envy-free item pricing in multi-item auctions, and (3) the level auctions of Morgenstern and Roughgarden [37] for single-item settings. The last result leads to an approximation of the overall optimal Myerson auction when bidders' valuations are drawn according to a fast-mixing Markov process, extending prior work that only gave such guarantees for the i.i.d. setting.Finally, we derive various extensions, including: (1) oracle-efficient algorithms for the contextual learning setting in which the learner has access to side information (such as bidder demographics), (2) learning with approximate oracles such as those based on Maximal-in-Range algorithms, and (3) noregret bidding in simultaneous auctions, resolving an open problem of Daskalakis and Syrgkanis [13].

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The k-center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case instances: a 2-approximation for symmetric k-center and an O(log * (k))-approximation for the asymmetric version. Therefore to improve on these ratios, one must go beyond the worst case.In this work, we take this approach and provide strong positive results both for the asymmetric and symmetric k-center problems under a natural input stability (promise) condition called α-perturbation resilience [Bilu and Linial, 2012], which states that the optimal solution does not change under any α-factor perturbation to the input distances. We provide algorithms that give strong guarantees simultaneously for stable and non-stable instances: our algorithms always inherit the worst-case guarantees of clustering approximation algorithms, and output the optimal solution if the input is 2-perturbation resilient. In particular, we show that if the input is only perturbation resilient on part of the data, our algorithm will return the optimal clusters from the region of the data that is perturbation resilient, while achieving the best worst-case approximation guarantee on the remainder of the data. Furthermore, we prove our result is tight by showing symmetric k-center under (2 − )-perturbation resilience is hard unless N P = RP .The impact of our results are multifaceted. First, to our knowledge, asymmetric k-center is the first problem that is hard to approximate to any constant factor in the worst case, yet can be optimally solved in polynomial time under perturbation resilience for a constant value of α. This is also the first tight result for any problem under perturbation resilience, i.e., this is the first time the exact value of α for which the problem switches from being NP-hard to efficiently computable has been found. Furthermore, our results illustrate a surprising relationship between symmetric and asymmetric k-center instances under perturbation resilience. Unlike approximation ratio, for which symmetric k-center is easily solved to a factor of 2 but asymmetric k-center cannot be approximated to any constant factor, both symmetric and asymmetric k-center can be solved optimally under resilience to 2-perturbations. Finally, our guarantees in the setting where only part of the data satisfies perturbation resilience makes these algorithms more applicable to real-life instances.

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by 1 σ times that of the uniform distribution; nature then samples an input from this distribution. Crucially, our results hold for adaptive adversaries that can base their choice of an input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm's current state; this appears to rule out the standard proof approaches in smoothed analysis.This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i.e., an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems: