High-dimensional shape fitting in linear time
Proceedings of the nineteenth annual symposium on Computational geometry
The NP-completeness column: The many limits on approximation
ACM Transactions on Algorithms (TALG)
Efficient subspace approximation algorithms
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
| Hi-index | 0.00 |
Let P be a set of n points in \mathbb{R}^d. For any 1 \leqslant k \leqslant d, the outer k-radius of P, denoted by Rk (P), is the minimum, over all (d - k)-dimensional flats F, of \max _{p\varepsilon P} d(p,f), where d (p, F) is the Euclidean distance between the pointp and flat F . We consider the scenario when the dimension d is not fixed and can be as large as n. Computing the various radii of point sets is a fundamental problem in computational convexity with many applications.The main result of this paper is a randomized polynomial time algorithm that approximates Rk(P) to within a factor of 0(\sqrt {\log n \cdot \log d}) for any 1 \leqslant k \leqslant d. This algorithm is obtained using techniques from semidefinite programming and dimension reduction. Previously, good approximation algorithms were known only for the case k =1and for the case when k = d - c for any constant c ; there are polynomial time algorithms that approximate Rk(P) to within a factor of (1 + \varepsilon), for any e0, when d - k is any fixed constant [23, 7]. On the other hand, some results from the mathematical programming community on approximating certain kinds of quadratic programs [28, 27] imply an 0(\sqrt {\log n} ) approximation for R1(P), the width of the pointset P.We also prove an inapproximability result for computing Rk(P), which easily yields the conclusion that our approximation algorithm performs quite well for a large range ofvalues of k . Our inapproximability result for Rk(P) improves the previous known hardness result of Brieden [13], and is proved by improving the parameters in Brieden's construction using basic ideas from PCP theory.