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 [15]. 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 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 [22, 7]. 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) Y. Ye and J. Zhang, An improved algorithm for approximating the radii of point sets, Proc. APPROX, 2003. Department of Computer Science, The University of Iowa, Iowa City, IA 52242-1419, email: [email protected], www: http://www.cs.uiowa.edu/ ̃kvaradar/ Department of Computer Science, University of Victoria, PO Box 3055, STN CSC, Victoria, BC, Canada V8W 3P6, email: [email protected], www: http://www.cs.uvic.ca/ ̃venkat Management Science and Engineering and, by courtesy, Electrical Engineering, Stanford University, Stanford, CA 94305, USA, email: [email protected] www: http://www.stanford.edu/ ̃yyye/ IOMS-Operations Management, Stern School of Business, New York University, 44 W. 4th Street, Suite 8-66, New York, NY 10012-1126, email: [email protected] www: www.stern.nyu.edu/ ̃jzhang