Bi-criteria linear-time approximations for generalized k-mean/median/center

We consider the problem of approximating a set P of n points in Rd by a collection of j-dimensional flats, and extensions thereof, under the standard median / mean / center measures, in which we wish to minimize, respectively, the sum of the distances from each point of P to its nearest flat, the sum of the squares of these distances, or the maximal such distance. Such problems cannot be approximated 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 bi-criteria approximation algorithm, which produces at most (k, j, n) = log n

• Type: Conference
• Year: 2007
• Where: COMPGEOM
• Authors: Dan Feldman, Amos Fiat, Micha Sharir, Danny Segev
• Keywords: Bi-criteria Approximation, COMPGEOM 2007, Discrete Geometry, J-dimensional Flats, Objective Functions