Abstract

We consider the problem of computing the outer‐radii of point sets. In this problem, we are given integers $n, d$, and k, where $k \le d$, and a set P of n points in $\Re^d$. The goal is to compute the outer k‐radius of 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.

MSC codes

  1. 68W20
  2. 68W25
  3. 68W40

Keywords

  1. approximation algorithms
  2. semidefinite programming
  3. computational convexity

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References

1.
D. Achlioptas and F. McSherry, Fast computation of low rank matrix approximations, in Proceedings of the ACM Symposium on the Theory of Computing, 2001.
2.
P. K. Agarwal, S. Har‐Peled, and K. R. Varadarajan, Approximating extent measures of points, J. ACM, 51 (2004), pp. 606–635.
3.
S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy, Proof verification and hardness of approximation problems, J. ACM, 45 (1998), pp. 501–555.
4.
S. Arora and S. Safra, Probabilistic checking of proofs: A new characterization of NP, J. ACM, 45 (1998), pp. 70–122.
5.
Y. Azar, A. Fiat, A. Karlin, F. McSherry, and J. Saia, Spectral analysis of data, in Proceedings of the ACM Symposium on the Theory of Computing, 2001.
6.
G. Barequet and S. Har‐Peled, Efficiently approximating the minimum‐volume bounding box of a point set in three dimensions, J. Algorithms, 38 (2001), pp. 91–109.
7.
M. Bădoiu, S. Har‐Peled, and P. Indyk, Approximate clustering via core‐sets, in Proceedings of the ACM Symposium on the Theory of Computing, 2002.
8.
M. Bellare and P. Rogaway, The complexity of approximating a nonlinear program, Math. Program. B, 69 (1995), pp. 429–441.
9.
D. Bertsimas and Y. Ye, Semidefinite Relaxations, Multivariate Normal Distributions, and Order Statistics, Handbook Combin. Optim. 3, D.‐Z. Du and P. M. Pardalos, eds., Kluwer Academic Publishers, Norwell, MA, 1998, pp. 1–19.
10.
H. L. Bodlaender, P. Gritzmann, V. Klee, and J. Van Leeuwen, The computational complexity of norm maximization, Combinatorica, 10 (1990), pp. 203–225.
11.
A. Brieden, P. Gritzmann, and V. Klee, Inapproximability of some geometric and quadratic optimization problems, in Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems, P. M. Pardalos, ed., Kluwer Academic Publishers, Norwell, MA, 2000, pp. 96–115.
12.
A. Brieden, P. Gritzmann, R. Kannan, V. Klee, L. Lovasz, and M. Simonovits, Deterministic and randomized polynomial‐time approximation of radii, Mathematika, 48 (2001), pp. 63–105.
13.
A. Brieden, P. Gritzmann, R. Kannan, V. Klee, L. Lovasz, and M. Simonovits, Approximation of diameters: Randomization doesn’t help, in Proceedings of the IEEE Symposium on the Foundations of Computer Science, 1998, pp. 244–251.
14.
A. Brieden, Geometric optimization problems likely not contained in APX, Discrete Comput. Geom., 28 (2002), pp. 201–209.
15.
U. Faigle, W. Kern, and M. Streng, Note on the computational complexity of j‐radii of polytopes in $R^n$, Math. Program., 73 (1996), pp. 1–5.
16.
U. Feige, A threshold of $\ln n$ for approximating set cover, J. ACM, 45 (1998), pp. 634–652.
17.
A. Frieze, R. Kannan, and S. Vempala, Fast Monte‐Carlo algorithms for finding low rank approximations, J. ACM, 51 (2004), pp. 1025–1041.
18.
M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semi‐definite programming, J. ACM, 42 (1995), pp. 1115–1145.
19.
P. Gritzmann and V. Klee, Inner and outer j‐radii of convex bodies in finite‐dimensional normed spaces, Discrete Comput. Geom., 7 (1992), pp. 255–280.
20.
P. Gritzmann and V. Klee, Computational complexity of inner and outer j‐radii of polytopes in finite‐dimensional normed spaces, Math. Program., 59 (1993), pp. 162–213.
21.
P. Gritzmann and V. Klee, On the complexity of some basic problems in computational convexity: I. Containment problems, Discrete Math., 136 (1994), pp. 129–174.
22.
S. Har‐Peled and K. Varadarajan, Projective clustering in high dimensions using core‐sets, in Proceedings of the 18th Annual Symposium on Computational Geometry, ACM Press, 2002, pp. 312–318.
23.
S. Har‐Peled and K. Varadarajan, High‐dimensional shape fitting in linear time, Discrete Comput. Geom., 32 (2004), pp. 269–288.
24.
J. Håstad, Some optimal inapproximability results, J. ACM, 48 (2001), pp. 798–859.
25.
N. Megiddo, On the complexity of some geometric problems in unbounded dimension, J. Symbolic Comput., 10 (1990), pp. 327–334.
26.
A. Nemirovski, C. Roos, and T. Terlaky, On maximization of quadratic forms over intersection of ellipsoids with common center, Math. Program., 86 (1999), pp. 463–473.
27.
Yu. Nesterov, Global quadratic optimization via conic relaxation, in Handbook of Semidefinite Programming Theory, Algorithms, and Applications, H. Wolkowicz, R. Saigal, and L. Vandenberghe, eds., Kluwer Academic Publishers, Norwell, MA, 2000.
28.
R. Raz, A parallel repetition theorem, SIAM J. Comput., 27 (1998), pp. 763–803.

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Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 1764 - 1776
ISSN (online): 1095-7111

History

Submitted: 24 March 2005
Accepted: 2 November 2006
Published online: 19 March 2007

MSC codes

  1. 68W20
  2. 68W25
  3. 68W40

Keywords

  1. approximation algorithms
  2. semidefinite programming
  3. computational convexity

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