Barrier Forming: Separating Polygonal Sets with Minimum Number of Lines

TL;DR

This paper investigates the problem of optimally placing the minimum number of straight line barriers to separate multiple disjoint object sets in the plane, with applications in sensor placement and security.

Contribution

It introduces new methods for solving barrier forming problems for point sets and polygonal obstacles, including optimal and approximate solutions, and demonstrates their effectiveness.

Findings

Optimal solutions for point set separation using mathematical programming.

Effective 2-OPT approximation for polygonal obstacle separation.

Demonstrated practical applicability through extensive simulations.

Abstract

In this work, we carry out structural and algorithmic studies of a problem of barrier forming: selecting theminimum number of straight line segments (barriers) that separate several sets of mutually disjoint objects in the plane. The problem models the optimal placement of line sensors (e.g., infrared laser beams) for isolating many types of regions in a pair-wise manner for practical purposes (e.g., guarding against intrusions). The problem is NP-hard even if we want to find the minimum number of lines to separate two sets of points in the plane. Under the umbrella problem of barrier forming with minimum number of line segments, three settings are examined: barrier forming for point sets, point sets with polygonal obstacles, polygonal sets with polygonal obstacles. We describe methods for computing the optimal solution for the first two settings with the assistance of mathematical programming, and provide a 2-OPT solution for the third. We demonstrate the effectiveness of our methods through extensive simulations.