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Multi-Covering a Point Set by $m$ Disks with Minimum Total Area

Mariem Guitouni, Chek-Manh Loi, Sándor P. Fekete, Michael Perk, Aaron T. Becker

TL;DR

A fast heuristic is provided and analyzed that is used to initialize an exact Integer Program-ming solution and enforce separation constraints between the sensors by modifying the integer program formulation and by changing the disk candidate set.

Abstract

A common robotics sensing problem is to place sensors to robustly monitor a set of assets, where robustness is assured by requiring asset $p$ to be monitored by at least $κ(p)$ sensors. Given $n$ assets that must be observed by $m$ sensors, each with a disk-shaped sensing region, where should the sensors be placed to minimize the total area observed? We provide and analyze a fast heuristic for this problem. We then use the heuristic to initialize an exact Integer Programming solution. Subsequently, we enforce separation constraints between the sensors by modifying the integer program formulation and by changing the disk candidate set.

Multi-Covering a Point Set by $m$ Disks with Minimum Total Area

TL;DR

A fast heuristic is provided and analyzed that is used to initialize an exact Integer Program-ming solution and enforce separation constraints between the sensors by modifying the integer program formulation and by changing the disk candidate set.

Abstract

A common robotics sensing problem is to place sensors to robustly monitor a set of assets, where robustness is assured by requiring asset to be monitored by at least sensors. Given assets that must be observed by sensors, each with a disk-shaped sensing region, where should the sensors be placed to minimize the total area observed? We provide and analyze a fast heuristic for this problem. We then use the heuristic to initialize an exact Integer Programming solution. Subsequently, we enforce separation constraints between the sensors by modifying the integer program formulation and by changing the disk candidate set.

Paper Structure

This paper contains 18 sections, 3 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Solving GMC: Lower Bounds
  4. GMC heuristic
  5. Integer Programming
  6. Computing the Candidate Set
  7. GMC IP Formulation
  8. Upper Bounds: Enforcing separation constraints
  9. Introducing Separation Constraints
  10. Modifying the Candidate Set
  11. Results
  12. GMC
  13. Comparison of runtime cost
  14. Comparison of total area cost
  15. DGMC
  16. ...and 3 more sections

Figures (7)

  • Figure 1: Ground-based Kilobot robots, commanded by overhead controllers via infrared communication rubenstein2012kilobot. A LED spotlight underneath each drone represents the communication link. 25 Kilobots are covered by at least one drone, but three high-value robots are multi-covered by two drones. To minimize energy consumption, it is desirable to make the coverage disks as small as possible.
  • Figure 2: Optimal (minimum total area) solutions for the GMC and DGMC with $n=10$, $m=5$, and $\kappa$ between $1$ and $3$. (Left) Without separation constraints, the sum of the areas is equal to 122.93m. Note that there is a radius $0$ disk on the leftmost point. (Right) Enforcing a minimum distance of $\ell=3$ yields an optimal solution with a total area of 140.66m. The gray lines indicate the distance between the disks.
  • Figure 3: Solutions from uni_lg for the DGMC with different separation constraints; $\ell=0,5,10$, $n=40$ and $m=30$.
  • Figure 4: Comparison of runtime, total area, and optimality gap between the GMC IP solver and the heuristic. On all plots, lower is better. (Left) uni_sm; fixed $m=20$ variable $n$ (Right) uni_fix_n; fixed $n=250$ variable $m$.
  • Figure 5: Influence of clique constraints on the runtime of the DGMC IP on uni_sm; $m=20$ and $\ell=5$.
  • ...and 2 more figures