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Person# Ashkan Norouzi Fard

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Algorithm

In mathematics and computer science, an algorithm (ˈælɡərɪðəm) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algo

Approximation algorithm

In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable

Ashkan Norouzi Fard, Ola Nils Anders Svensson, Justin Dean Ward

Clustering is a classic topic in optimization with k-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best-known algorithm for k-means in Euclidean space with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of 9+epsilon, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of k-means and (2) satisfy the hard constraint that at most k clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard linear programming (LP) relaxation. Our techniques are quite general, and we also show improved guarantees for k-median in Euclidean metrics and for a generalization of k-means in which the underlying metric is not required to be Euclidean.

Moran Feldman, Ashkan Norouzi Fard, Ola Nils Anders Svensson, Rico Zenklusen

We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multi-player model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the above-mentioned connections, have multiple implications in the data stream and robust setting. Even for just two players, a prior information-theoretic hardness result implies that no approximation factor above 1/2 can be achieved in our model, if only queries to feasible sets, i.e., sets respecting the cardinality constraint, are allowed. We show that the possibility of querying infeasible sets can actually be exploited to beat this bound, by presenting a tight 2/3-approximation taking exponential time, and an efficient 0.514-approximation. To the best of our knowledge, this is the first example where querying a submodular function on infeasible sets leads to provably better results. Through the above-mentioned link to the robust setting, both of these algorithms improve on the current state-of-the-art for robust submodular maximization, showing that approximation factors beyond 1/2 are possible. Moreover, exploiting the link of our model to streaming, we settle the approximability for streaming algorithms by presenting a tight 1/2 + epsilon hardness result, based on the construction of a new family of coverage functions. This improves on a prior 1 - 1/e + epsilon hardness and matches, up to an arbitrarily small margin, the best known approximation algorithm.

Mikhail Kapralov, Slobodan Mitrovic, Ashkan Norouzi Fard, Jakab Tardos

Given a source of iid samples of edges of an input graph G with n vertices and m edges, how many samples does one need to compute a constant factor approximation to the maximum matching size in G? Moreover, is it possible to obtain such an estimate in a small amount of space? We show that this problem cannot be solved using a nontrivially sublinear (in m) number of samples: m(1-o(1)) samples are needed. On the other hand, O(log(2) n) bits of space suffice to compute an estimate. Our main technical tool is a new peeling type algorithm for matching and its local simulation. We show that a delicate balance between exploration depth and sampling rate allows our simulation to not lose precision over a logarithmic number of levels of recursion and achieve a constant factor approximation. Our algorithm also yields a constant factor approximate local computation algorithm (LCA) for matching with O(d log n) exploration starting from any vertex. Previous approaches were based on local simulations of randomized greedy, which take O(d) time in expectation over the starting vertex or edge (Yoshida et al'09, Onak et al'12), and could not achieve a better than d(2) runtime. Interestingly, we also show that unlike our algorithm, the local simulation of randomized greedy that is the basis of the most efficient prior results does take (Omega) over tilde (d(2)) >> O(d log n) time for a worst case edge even for d = exp(Theta(root log n)).