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In this paper we present improved bounds for approximating maximum matchings in bipartite graphs in the streaming model. First, we consider the question of how well maximum matching can be approximated in a single pass over the input whenÕ(n) space is allowed, where n is the number of vertices in the input graph. Two natural variants of this problem have been considered in the literature: (1) the edge arrival setting, where edges arrive in the stream and (2) the vertex arrival setting, where vertices on one side of the graph arrive in the stream together with all their incident edges. The latter setting has also been studied extensively in the context of online algorithms, where each arriving vertex has to either be matched irrevocably or discarded upon arrival. In the online setting, the celebrated algorithm of Karp-Vazirani-Vazirani achieves a 1 − 1/e approximation by crucially using randomization (and usingÕ(n) space). Despite the fact that the streaming model is less restrictive in that the algorithm is not constrained to match vertices irrevocably upon arrival, the best known approximation in the streaming model with vertex arrivals andÕ(n) space is the same factor of 1 − 1/e. We show that no (possibly randomized) single pass streaming algorithm constrained to useÕ(n) space can achieve a better than 1 − 1/e approximation to maximum matching, even in the vertex arrival setting. This leads to the striking conclusion that no single pass streaming algorithm can get any advantage over online algorithms unless it uses significantly more thanÕ(n) space. Additionally, our bound yields the best known impossibility result for approximating matchings in the edge arrival model (improving upon the bound of 2/3 proved by Goel at al [SODA'12]).Second, we consider the problem of approximating matchings in multiple passes in the vertex arrival setting. We show that a simple fractional load balancing approach achieves approximation ratio 1 − e −k k k−1 /(k − 1)! = 1 − 1 √ 2πk + o(1/k) in k passes using linear space. Thus, our algorithm achieves the best possible 1 − 1/e approximation in a single pass and improves upon the 1 − O( log log k/k) approximation in k passes due to Ahn and Guha [ICALP'11]. Additionally, our approach yields an efficient solution to the Gap-Existence problem considered by Charles et al [EC'10]. 0 side to the set of impressions [3]. An important constraint is that the set of impressions Q may be so large that it is not feasible to represent it explicitly, ruling out algorithms that take O(|P | + |Q|) space.Data model for lop-sided graphs. Since the set Q cannot be represented explicitly, it is important to fix the model of access to Q. Here we assume the following scenario. Vertices in P arrive in the stream in an adversarial order, together with a representation of their edges. We make no assumptions on the way the edges are represented. For example, some edges could be stored explicitly, while others may be represented implicitly. We assume access to the following two functions:1. LIST-NEIGHBORS(u, S)...

Low rank approximation is a fundamental computational primitive widely used in data analysis. In many applications the dataset that the algorithm operates on may contain sensitive information about contributing individuals (e.g. user/movie ratings in the Netflix challenge), motivating the need to design low rank approximation algorithms that preserve privacy of individual entries of the input matrix.In this paper, we give a polynomial time algorithm that, given a privacy parameter > 0, for a symmetric matrix A, outputs an -differentially approximation to the principal eigenvector of A, and then show how this algorithm can be used to obtain a differentially private rank-k approximation. We also provide lower bounds showing that our utility/privacy tradeoff is close to best possible.While there has been significant progress on this problem recently for a weaker notion of privacy, namely ( , δ)-differential privacy [HR12, BBDS12], our result is the first to achieve ( , 0)-differential privacy guarantees with a near-optimal utility/privacy tradeoff in polynomial time.

We present a streaming algorithm that makes one pass over the edges of an unweighted graph presented in random order, and produces a polylogarithmic approximation to the size of the maximum matching in the graph, while using only polylogarithmic space. Prior to this work the only approximations known were a folkloreÕ( √ n) approximation with polylogarithmic space in an n vertex graph and a constant approximation with Ω(n) space. Our work thus gives the first algorithm where both the space and approximation factors are smaller than any polynomial in n.Our algorithm is obtained by effecting a streaming implementation of a simple "local" algorithm that we design for this problem. The local algorithm produces a O(k · n 1/k ) approximation to the size of a maximum matching by exploring the radius k neighborhoods of vertices, for any parameter k. We show, somewhat surprisingly, that our local algorithm can be implemented in the streaming setting even for k = Ω(log n/ log log n). Our analysis exposes some of the problems that arise in such conversions of local algorithms into streaming ones, and gives techniques to overcome such problems.

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