TEL-AVIV UNIVERSITY RAYMOND AND BEVERLY SACKLER FACULTY OF EXACT SCIENCES BLAVATNIK SCHOOL OF COMPUTER SCIENCE Coresets and Their Applications

In this thesis we investigate the construction and applications of coresets (small sets which approximately represent much larger input sets, in term of various objective measures) to several problems in geometric optimization. Bi-criteria approximation algorithms We consider the problem of approximating a set P of n points inR by a collection of k j-dimensional flats, and extensions thereof, under the median / mean / center measures, in which we wish to minimize, respectively, the sum of the Euclidean distances from each point of P to its nearest flat, the sum of the squares of these distances, and the maximum such distance. Problems of this kind belong to the area of projective clustering. Such problems cannot be approximated in polynomial time, for every approximation factor, unless P=NP but do allow bi-criteria approximations, where one allows some leeway in both the number of flats and the quality of the objective function. We give a very simple randomized bi-criteria approximation algorithm, which produces, with high probability, at most α(k, j, n) = log n · (jk log log n) flats, which exceeds the optimal objective value for any k jdimensional flats by a factor of no more than β(j) = 2. We use this bi-criteria approximation in the construction of coresets for projective clustering; see Chapter 4. Our bi-criteria algorithm has many advantages over previous work, in that it is much more widely applicable (wider set of objective functions and classes of clusters) and much more efficient — reducing the running time bound from O(n) to O(dn) · (jk). We give full details of this work in Chapter 3. A preliminary version has appeared in [FFSS07]; Since the publication of [FFSS07] in 2007 it has been cited and used in subsequent work [FL08, FFKN09, FMSW10].