We consider the problem of computing the outer-radii of point sets. In this problem, we are given integers n, d, k where k ≤ d, and a set P of n points in R d . The goal is to compute the outer k-radius of P , denoted by R k (P ), which is the minimum, over all is the Euclidean distance between the point p and flat F . Computing the radii of point sets is a fundamental problem in computational convexity with many significant applications. The problem admits a polynomial time algorithm when the dimension d is constant [16] . Here we are interested in the general case when the dimension d is not fixed and can be as large as n, where the problem becomes NP-hard even for k = 1. It is known that R k (P ) can be approximated in polynomial time by a factor of (1 + ε), for any ε > 0, when d − k is a fixed constant [8, 23] . A * Research by the first author is supported by NSF CAREER award CCR-0237431. Research by the second author is supported by an NSERC discovery grant. Research by the third and fourth authors is supported by NSF grant DMI-0231600. A preliminary version of this paper appeared as : (i) K. R. Varadarajan, S. Venkatesh, and J. Zhang, Approximating the radii of point sets in high dimensions, Proc. 43rd IEEE Symp. Found. Comput. Sci., 2002, and (ii) 1 polynomial time algorithm that guarantees a factor of O( √ log n) approximation for R 1 (P ), the width of the point set P , is implied by the results of Nemirovski et al. [29] and Nesterov [28] . In this paper, we show that R k (P ) can be approximated by a ratio of O( √ log n) for any 1 ≤ k ≤ d, thus matching the previously best known ratio for approximating the special case R 1 (P ), the width of point set P . Our algorithm is based on semidefinite programming relaxation with a new mixed deterministic and randomized rounding procedure. We also prove an inapproximability result that gives evidence that our approximation algorithm is doing well for a large range of k. We show that there exists a constant δ > 0 such that the following holds for any 0 < ε < 1: there is no polynomial time algorithm that approximates R k (P ) within (log n) Our inapproximability result for R k (P ) extends a previously known hardness result of Brieden [15], and is proved by modifying Brieden's construction using basic ideas from PCP theory.
In fact, there is a constant δ > 0 such that it is Quasi-NP-hard, i.e., implies NP ⊆ DTIME(2 (log n) O(1) ), to even find a (log n) δ approximation to the best fitting k-subspace when k ≥ d for any fixed > 0 REF .
On Approximating the Radii of Point Sets in High Dimensions
FOCS
Mathematics Computer Science