An Improved Algorithm for Approximating the Radii of Point Sets

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. 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 Nemirovski [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 factor 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.