Covering a line segment with variable radius discs

TL;DR

This paper studies the problem of optimally placing sensors with variable-radius coverage to fully surveil a line segment, proposing algorithms for both single and multiple sensor types, with practical heuristics showing high effectiveness.

Contribution

It introduces algorithms for the line segment covering problem with variable-radius sensors, including a polynomial solution for single sensor types and heuristics for multiple types.

Findings

The polynomial algorithm efficiently solves the single sensor type case.

The heuristic often finds the optimal solution in computational tests.

The problem is NP-hard when multiple sensor types are involved.

Abstract

The paper addresses the problem of locating sensors with a circular field of view so that a given line segment is under full surveillance, which is termed as the Disc Covering Problem on a Line. The cost of each sensor includes a fixed component, and a variable component that is proportional to the field-of-view area. When only one type of sensor or, in general, one type of disc, is available, then a simple polynomial algorithm solves the problem. When there are different types of sensors in terms of fixed and variable costs, the problem becomes NP-hard. A branch-and-bound algorithm as well as an efficient heuristic are developed. The heuristic very often obtains the optimal solution as shown in extensive computational testing.