An improved algorithm for approximating the radii of point sets

Abstract

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 Rd. The goal is to compute the outer k-radius of P, denoted by Rk(P), which is the minimum, over all (d - k)-dimensional flats F, of maxp∈P d(p, F), where d(p, F) 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 significantly many applications. The problem admits a polynomial time algorithm when the dimension d is constant [9]. 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 has been known that Rk(P) can be approximated in polynomial time by a factor of (1+ε), for any ε > 0, when d - k is a fixed constant [15,2]. A factor of O(√log n) approximation for R1(P), the width of the point set P, is implied from the results of Nemirovskii et al. [19] and Nesterov [18]. The first approximation algorithm for general k has been proposed by Varadarajan, Venkatesh and Zhang [20]. Their algorithm is based on semidefinite programming relaxation and the Johnson-Lindenstrauss lemma, and it has a performance guarantee of O(√log n · log d). In this paper, we show that Rk(P) can be approximated by a ratio of O(√/log n) for any 1 ≤ k ≤ d and thereby improve the ratio of [20] by a factor of O(√log d) that could be as large as O(√log n). This ratio also matches the previously best known ratio for approximating the special case R1(P), the width of point set P. Our algorithm is based on semidefinite programming relaxation with a new mixed deterministic and randomized rounding procedure. © Springer-Verlag Berlin Heidelberg 2003.