Approximation Algorithms for a Triangle Enclosure Problem.

Given a set S of n points in the plane, we want to nd a triangle, with vertices in S, such that the number of points enclosed by it is maximum. An exact solution can be found by considering all n triples of points in S. We consider the problem of approximating this triangle. We show that, by considering only triangles with at least 1, 2, or 3 vertices on the convex hull of S, we obtain various approximation algorithms that run in o(n 3 ) time.