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The Lovász Local Lemma [5] (LLL) is a powerful result in probability theory that states that the probability that none of a set of bad events happens is nonzero if the probability of each event is small compared to the number of events that depend on it. It is often used in combination with the probabilistic method for non-constructive existence proofs. A… (More)

Star-shaped bodies are an important nonconvex generalization of convex bodies (e.g., linear programming with violations). Here we present an efficient algorithm for sampling a given star-shaped body. The complexity of the algorithm grows polynomially in the dimension and inverse polynomi-ally in the fraction of the volume taken up by the kernel of the… (More)

We study the problem of learning the most biased coin among a set of coins by tossing the coins adaptively. The goal is to minimize the number of tosses to identify a coin i * such that Pr (coin i * is most biased) is at least 1 − δ for any given δ. Under a particular probabilistic model, we give an optimal algorithm, i.e., an algorithm that minimizes the… (More)

- Salil Vadhan, Aleksandar Makelov, Karthekeyan Chandrasekaran, Ameya Velingker, Curtis McMullen, Alexandra Kolla +3 others
- 2015

Acknowledgements First, I'd like to thank my family for their love and encouragement, without which I wouldn't have been able to follow my passion for mathematics. Any success I've had is rooted in the eeort they invested in my upbringing. Next, this thesis would not have been possible without the support of my advisor, Professor Salil Vadhan, who pointed… (More)

A hitting set for a collection of sets is a set that has a non-empty intersection with each set in the collection; the hitting set problem is to find a hitting set of minimum cardinality. Motivated by instances of the hitting set problem where the number of sets to be hit is large, we introduce the notion of <i>implicit hitting set problems</i>. In an… (More)

We study the problem of answering <i>k</i>-way marginal queries on a database <i>D</i> ϵ ({0,1}<sup>d</sup>)<sup>n</sup>, while preserving differential privacy. The answer to a <i>k</i>-way marginal query is the fraction of the database's records <i>x</i> in {0,1}<sup>d</sup> with a given value in each of a given set of up to <i>k</i> columns.… (More)

An undirected graph G = (V, E) is stable if its inessential vertices (those that are exposed by at least one maximum matching) form a stable set. We call a set of edges F ⊆ E a stabilizer if its removal from G yields a stable graph. In this paper we study the following natural edge-deletion question: given a graph G = (V, E), can we find a… (More)

The cutting plane approach to optimal matchings has been discussed by several authors over the past decades [21, 14, 17, 25, 10], and its convergence has been an open question. We prove that the cutting plane approach using Edmonds' blossom inequalities converges in polynomial time for the minimum-cost perfect matching problem. Our main insight is an… (More)

Efficient sampling, integration and optimization algorithms for logconcave functions [BV04,KV06,LV06a] rely on the good isoperime-try of these functions. We extend this to show that −1/(n − 1)-concave functions have good isoperimetry, and moreover, using a characterization of functions based on their values along every line, we prove that this is the… (More)