Low-Dimensional Linear Programming with Violations

Timothy M. Chan
2005 SIAM journal on computing (Print)
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 violations: finding a point inside all but at most of Ò given halfspaces. We give a simple algorithm in 2-d that runs in Ç´´Ò · ¾ µÐÓ Òµ expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matoušek (1994) and is probably
more » ... nearoptimal for all Ò ¾. A (theoretical) extension of our algorithm in 3-d runs in near Ç´Ò · ½½ Ò ½ µ expected time. Interestingly, the idea is based on concave-chain decompositions (or covers) of the´ µ-level, previously used in proving combinatorial -level bounds. Applications in the plane include improved algorithms for finding a line that misclassifies the fewest among a set of bichromatic points, and finding the smallest circle enclosing all but points. We also discuss related problems of finding local minima in levels. LP with violations: the problem and background. We can define the problem of LP with at most violations in dimensions as follows: Given a set À of Ò halfspaces in IR , a linear objective function , and an integer ¼ Ò ¾, minimize over the region Á ´Àµ Õ ¾ IR Õ lies outside halfspaces of À or report that Á ´Àµ . Since our interest is in geometric applications, we confine our discussion to small constant values of . (The problem for arbitrary dimensions is NP-complete; see for example the references at [22] under the name "Maximum Hyperplane Consistency.") Note that Á ¼´À µ, the intersection of all halfspaces, is the feasible region of the original LP problem. Following Matoušek [39], we call the special case of the problem in which Á ¼´À µ the feasible case. In the feasible case (where by