Separating Geometric Data with Minimum Cost: Two Disjoint Convex Hulls

Abstract

In this study, a geometric version of an NP-hard problem ("Almost $2-SAT$" problem) is introduced which has potential applications in clustering, separation axis, binary sensor networks, shape separation, image processing, etc. Furthermore, it has been illustrated that the new problem known as "Two Disjoint Convex Hulls" can be solved in polynomial time due to some combinatorial aspects and geometric properties. For this purpose, an $O(n^2)$ algorithm has also been presented which employs the Separating Axis Theorem (SAT) and the duality of points/lines.