Separating Geometric Data with Minimum Cost: Two Disjoint Convex Hulls

TL;DR

This paper introduces a polynomial-time algorithm for separating two disjoint convex hulls, a geometric problem related to NP-hard problems, with applications in various fields like image processing and sensor networks.

Contribution

It presents a new geometric problem called 'Two Disjoint Convex Hulls' and provides an $O(n^2)$ algorithm using the Separating Axis Theorem and duality principles.

Findings

The problem can be solved in polynomial time.

An $O(n^2)$ algorithm is developed for the separation task.

The approach leverages geometric properties and the Separating Axis Theorem.

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.