Given a set of points P ⊂ d and a kernel k, the Kernel Density Estimate at a point x ∈ d is defined as KDE P (x) 1 |P| y∈P k(x, y). We study the problem of designing a data structure that given a data set P and a kernel function, returns approximations to the kernel density of a query point in sublinear time. We introduce a class of unbiased estimators for kernel density implemented through locality-sensitive hashing, and give general theorems bounding the variance of such estimators. These estimators give rise to efficient data structures for estimating the kernel density in high dimensions for a variety of commonly used kernels. Our work is the first to provide data-structures with theoretical guarantees that improve upon simple random sampling in high dimensions.

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We study the problem of revenue maximization in the marketing model for social networks introduced by (Hartline, Mirrokni, Sundararajan, WWW '08). In this setting, a digital product is sold to a set of potential buyers under positive externalities, and the seller seeks for a marketing strategy, namely an ordering in which he approaches the buyers and the prices offered to them, that maximizes his revenue. We restrict our attention to the Uniform Additive Model and mostly focus on Influence-and-Exploit (IE) marketing strategies. We obtain a comprehensive collection of results on the efficiency and the approximability of IE strategies, which also imply a significant improvement on the best known approximation ratios for revenue maximization. Specifically, we show that in the Uniform Additive Model, both computing the optimal marketing strategy and computing the best IE strategy are NP-hard for undirected social networks. We observe that allowing IE strategies to offer prices smaller than the myopic price in the exploit step leads to a measurable improvement on their performance. Thus, we show that the best IE strategy approximates the maximum revenue within a factor of 0.911 for undirected and of roughly 0.553 for directed networks. Moreover, we present a natural generalization of IE strategies, with more than two pricing classes, and show that they approximate the maximum revenue within a factor of roughly 0.7 for undirected and of roughly 0.35 for directed networks. Utilizing a connection between good IE strategies and large cuts in the underlying social network, we obtain polynomial-time algorithms that approximate the revenue of the best IE strategy within a factor of roughly 0.9. Hence, we significantly improve on the best known approximation ratio for revenue maximization to 0.8229 for undirected and to 0.5011 for directed networks (from 2/3 and 1/3, respectively, by Hartline et al.).

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In the study of random structures we often face a trade-off between realism and tractability, the latter typically enabled by assuming some form of independence. In this work we initiate an effort to bridge this gap by developing tools that allow us to work with independence without assuming it. Let Gn be the set of all graphs on n vertices and let S be an arbitrary subset of Gn, e.g., the set of graphs with m edges. The study of random networks can be seen as the study of properties that are true for most elements of S, i.e., that are true with high probability for a uniformly random element of S. With this in mind, we pursue the following question: What are general sufficient conditions for the uniform measure on a set of graphs S ⊆ Gn to be approximable by a product measure?