ABSTRACT:A result of Johnson and Lindenstrauss [13] shows that a set of n points in high dimensional Euclidean space can be mapped into an O(log n/⑀ 2 )-dimensional Euclidean space such that the distance between any two points changes by only a factor of (1 Ϯ ⑀). In this note, we prove this theorem using elementary probabilistic techniques.

The doubling constant of a metric space´ µ is the smallest value such that every ball in can be covered by balls of half the radius. The doubling dimension of is then defined as Ñ´ µ ÐÓ ¾ . A metric (or sequence of metrics) is called doubling precisely when its doubling dimension is bounded. This is a robust class of metric spaces which contains many families of metrics that occur in applied settings. We give tight bounds for embedding doubling metrics into (low-dimensional) normed spaces. We consider both general doubling metrics, as well as more restricted families such as those arising from trees, from graphs excluding a fixed minor, and from snowflaked metrics. Our techniques include decomposition theorems for doubling metrics, and an analysis of a fractal in the plane due to Laakso [21]. Finally, we discuss some applications and point out a central open question regarding dimensionality reduction in Ä ¾ .

Consider the following problem: given a metric space, some of whose points are "clients," select a set of at most k facility locations to minimize the average distance from the clients to their nearest facility. This is just the well-studied k-median problem, for which many approximation algorithms and hardness results are known. Note that the objective function encourages opening facilities in areas where there are many clients, and given a solution, it is often possible to get a good idea of where the clients are located. This raises the following quandary: what if the locations of the clients are sensitive information that we would like to keep private? Is it even possible to design good algorithms for this problem that preserve the privacy of the clients?In this paper, we initiate a systematic study of algorithms for discrete optimization problems in the framework of differential privacy (which formalizes the idea of protecting the privacy of individual input elements). We show that many such problems indeed have good approximation algorithms that preserve differential privacy; this is even in cases where it is impossible to preserve cryptographic definitions of privacy while computing any non-trivial approximation to even the value of an optimal solution, let alone the entire solution.Apart from the k-median problem, we consider the problems of vertex and set cover, min-cut, k-median, facility location, and Steiner tree, and give approximation algorithms and lower bounds for these problems. We also consider the recently introduced submodular maximization problem, "Combinatorial Public Projects" (CPP), shown by Papadimitriou et al. [28] to be inapproximable to subpolynomial multiplicative factors by any efficient and truthful algorithm. We give a differentially private (and hence approximately truthful) algorithm that achieves a logarithmic additive approximation.

Consider a random graph model where each possible edge e is present independently with some probability p e . Given these probabilities, we want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, we are forced to add it to our matching. Further, each vertex i is allowed to be queried at most t i times. How should we adaptively query the edges to maximize the expected weight of the matching? We consider several matching problems in this general framework (some of which arise in kidney exchanges and online dating, and others arise in modeling online advertisements); we give LP-rounding based constant-factor approximation algorithms for these problems. Our main results are the following:• We give a 4 approximation for weighted stochastic matching on general graphs, and a 3 approximation on bipartite graphs. This answers an open question from [Chen et al. ICALP 09]. • Combining our LP-rounding algorithm with the natural greedy algorithm, we give an improved 3.46 approximation for unweighted stochastic matching on general graphs. • We introduce a generalization of the stochastic online matching problem [Feldman et al. FOCS 09] that also models preference-uncertainty and timeouts of buyers, and give a constant factor approximation algorithm.1 of unsuccessful dates a person might be willing to participate in, "timeouts" on vertices are also provided. More precisely, valid policies are allowed, for each vertex i, to only probe at most t i edges incident to i. Similar considerations arise in kidney exchanges, details of which appear in [7]. Chen et al. asked the question: how can we devise probing policies to maximize the expected cardinality (or weight) of the matching? They showed that the greedy algorithm that probes edges in decreasing order of p ij (as long as their endpoints had not timed out) was a 4-approximation to the cardinality version of the stochastic matching problem. This greedy algorithm (and other simple greedy schemes) can be seen to be arbitrarily bad in the presence of weights, and they left open the question of obtaining good algorithms to maximize the expected weight of the matching produced. In addition to being a natural generalization, weights can be used as a proxy for revenue generated in matchmaking services. (The unweighted case can be thought of as maximizing the social welfare.) In this paper, we resolve the main open question from Chen et al. [7]:Theorem 1 There is a 4-approximation algorithm for the weighted stochastic matching problem. For bipartite graphs, there is a 3-approximation algorithm.Our main idea is to use the knowledge of edge probabilities to solve a linear program where each edge e has a variable 0 ≤ y e ≤ 1 corresponding to the probability that a strategy probes e (over all possible realizations of the graph). This is similar to the approach for stochastic packing problems considered by Dean et al. [9,8]. We then give two d...

We give simple and easy-to-analyze randomized approximation algorithms for several well-studied NP-hard network design problems. Our algorithms improve over the previously best known approximation ratios. Our main results are the following.We give a randomized 3.55-approximation algorithm for the connected facility location problem. The algorithm requires three lines to state, one page to analyze, and improves the best-known performance guarantee for the problem.We give a 5.55-approximation algorithm for virtual private network design. Previously, constant-factor approximation algorithms were known only for special cases of this problem.We give a simple constant-factor approximation algorithm for the single-sink buy-at-bulk network design problem. Our performance guarantee improves over what was previously known, and is an order of magnitude improvement over previous combinatorial approximation algorithms for the problem.