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 R ^d. The goal is to compute the outer k -radius of P, denoted by R _k (P), which is the minimum, over all (d-k)-dimensional flats F, of max _p ∈ 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 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 [15,2]. A factor of $O(\sqrt{\log n})$ approximation for R ₁(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 $O(\sqrt{\log n \cdot \log d})$.

In this paper, we show that R _k (P) can be approximated by a ratio of $O(\sqrt{\log n})$ for any 1 ≤ k ≤ d and thereby improve the ratio of [20] by a factor of $O(\sqrt{\log d})$ that could be as large as $O(\sqrt{\log n})$. This ratio also matches 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.

Research supported in part by NSF grant DMI-0231600.