# Approximating the Radii of Point Sets

## Abstract

*k*, where $k \le d$, and a set

*P*of

*n*points in $\Re^d$. The goal is to compute the

*outer*of

*k*‐radius*P*, denoted by ${\cal R}_k(P)$, which is the minimum over all $(d-k)$‐dimensional flats

*F*of $\max_{p \in P} d(p,F)$, 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 many significant applications. The problem admits a polynomial time algorithm when the dimension

*d*is constant [U. Faigle, W. Kern, and M. Streng,

*Math. Program.*, 73 (1996), pp. 1–5]. Here we are interested in the general case in which 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 $(1 + \varepsilon)$ for any $\varepsilon > 0$ when $d - k$ is a fixed constant [M. Bădoiu, S. Har‐Peled, and P. Indyk, in

*Proceedings of the ACM Symposium on the Theory of Computing*, 2002; S. Har‐Peled and K. Varadarajan, in

*Proceedings of the ACM Symposium on Computing Geometry*, 2002]. A polynomial time algorithm that guarantees a factor of $O(\sqrt{\log n})$ approximation for $R_1(P)$, the width of the point set

*P*, is implied by the results of Nemirovski, Roos, and Terlaky [

*Math. Program.*, 86 (1999), pp. 463–473] and Nesterov [

*Handbook of Semidefinite Programming Theory, Algorithms*, Kluwer Academic Publishers, Norwell, MA, 2000]. In this paper, we show that $R_k(P)$ can be approximated by a ratio of $O(\sqrt{\log n})$ for any $1 \leq k \leq d$, 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 $\delta > 0$ such that the following holds for any $0 < \eps < 1$: there is no polynomial time algorithm that approximates $R_k(P)$ within $(\log n)^{\delta}$ for all

*k*such that $k \leq d - d^{\varepsilon}$ unless NP $\subseteq$ DTIME $[2^{(\log m)^{O(1)}}]$. Our inapproximability result for $R_k(P)$ extends a previously known hardness result of Brieden [

*Discrete Comput. Geom.*, 28 (2002), pp. 201–209] and is proved by modifying Brieden’s construction using basic ideas from probabilistically checkable proofs (PCP) theory.

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**Submitted**: 24 March 2005

**Accepted**: 2 November 2006

**Published online**: 19 March 2007

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