# On geometric optimization with few violated constraints

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DOI: 10.1007/BF02570713

- Cite this article as:
- Matoušek, J. Discrete & Computational Geometry (1995) 14: 365. doi:10.1007/BF02570713

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## Abstract

We investigate the problem of finding the best solution satisfying all but*k* of the given constraints, for an abstract class of optimization problems introduced by Sharir and Welzl—the so-called**LP**-type problems. We give a general algorithm and discuss its efficient implementations for specific geometric problems. For instance for the problem of computing the smallest circle enclosing all but*k* of the given*n* points in the plane, we obtain an*O*(*n* log*n*+*k*^{3}*n*^{ε}) algorithm; this improves previous results for*k* small compared with*n* but moderately growing. We also establish some results concerning general properties of**LP**-type problems.

## Copyright information

© Springer-Verlag New York Inc. 1995