Low-dimensional linear programming with violations

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 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 µ Ð Ó Òµ expected time; this is faster than earlier algorithms by Everett, Robert, and van Kreveld (1993) and Matoušek (1994) and is probably 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

Shape fitting with outliers

“…To a certain extent, the problem of handling outliers in computational geometry is still open, see [8,14,21] for relevant results. While those results provide relatively efficient solutions, most of them are restricted to two and three dimensions (where intuitively, the k-level has relatively low complexity), and are restricted in the type of problems that they can solve.…”

“…Namely, we compute a subset C of the points of cardinality k/ε O (1) , such that instead of solving the problem on the original point set, one can solve it on C. In particular, in some cases, we just plug in the (exact) algorithms of [8,12,14,21] on C, and get an efficient approximation algorithm.…”