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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)

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)

Stabilization of graphs has received substantial attention in recent years due to its connection to game theory. Stable graphs are exactly the graphs inducing a matching game with non-empty core. They are also the graphs that induce a network bargaining game with a balanced solution. A graph with weighted edges is called stable if the maximum weight of an… (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 polynomially in the fraction of the volume taken up by the kernel of the… (More)