## Low-Dimensional Linear Programming with Violations (2002)

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Venue: | In Proc. 43th Annu. IEEE Sympos. Found. Comput. Sci |

Citations: | 45 - 3 self |

### BibTeX

@INPROCEEDINGS{Chan02low-dimensionallinear,

author = {Timothy M. Chan},

title = {Low-Dimensional Linear Programming with Violations},

booktitle = {In Proc. 43th Annu. IEEE Sympos. Found. Comput. Sci},

year = {2002},

pages = {570--579}

}

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### Abstract

Two decades ago, Megiddo and Dyer showed that linear programming in 2 and 3 dimensions (and subsequently, any constant number of dimensions) can be solved in linear time. In this paper, we consider linear programming with at most k violations: finding a point inside all but at most k of n given halfspaces. We give a simple algorithm in 2-d that runs in O((n + k ) log n) expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matousek (1994) and is probably nearoptimal for all k n=2. A (theoretical) extension of our algorithm in 3-d runs in near O(n + k ) expected time. Interestingly, the idea is based on concave-chain decompositions (or covers) of the ( k)-level, previously used in proving combinatorial k-level bounds.