Advertisement

Min-Max Latency Walks: Approximation Algorithms for Monitoring Vertex-Weighted Graphs

  • Soroush Alamdari
  • Elaheh Fata
  • Stephen L. Smith
Part of the Springer Tracts in Advanced Robotics book series (STAR, volume 86)

Abstract

In this paper, we consider the problem of planning a path for a robot to monitor a known set of features of interest in an environment.We represent the environment as a vertex- and edge-weighted graph, where vertices represent features or regions of interest. The edge weights give travel times between regions, and the vertex weights give the importance of each region. If the robot repeatedly performs a closed walk on the graph, then we can define the latency of a vertex to be the maximum time between visits to that vertex, weighted by the importance (vertex weight) of that vertex. Our goal in this paper is to find the closed walk that minimizes the maximum weighted latency of any vertex. We show that there does not always exist an optimal walk of polynomial size. We then prove that for any graph there exist a constant approximation walk of size O(n 2), where n is the number of vertices. We provide two approximation algorithms; an O(log n)-approximation and an O(log ρ)-approximation, where ρ is the ratio between the maximum and minimum vertex weight. We provide simulation results which demonstrate that our algorithms can be applied to problems consisting of thousands of vertices.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bullo, F., Frazzoli, E., Pavone, M., Savla, K., Smith, S.L.: Dynamic vehicle routing for robotic systems. Proceedings of the IEEE 99(9), 1482–1504 (2011)CrossRefGoogle Scholar
  2. 2.
    Caffarelli, L., Crespi, V., Cybenko, G., Gamba, I., Rus, D.: Stochastic distributed algorithms for target surveillance. In: Intelligent Systems and Design Applications, Tulsa, OK, pp. 137–148 (2003)Google Scholar
  3. 3.
    Cannata, G., Sgorbissa, A.: A minimalist algorithm for multirobot continuous coverage. IEEE Transactions on Robotics 27(2), 297–312 (2011)CrossRefGoogle Scholar
  4. 4.
    Chevaleyre, Y.: Theoretical analysis of the multi-agent patrolling problem. In: IEEE/WIC/ACM Int. Conf. Intelligent Agent Technology, Beijing, China, pp. 302–308 (2004)Google Scholar
  5. 5.
    Choset, H.: Coverage for robotics – A survey of recent results. Annals of Mathematics and Artificial Intelligence 31(1-4), 113–126 (2001)CrossRefGoogle Scholar
  6. 6.
    Christofides, N., Beasley, J.E.: The period routing problem. Networks 14(2), 237–256 (1984)CrossRefMATHGoogle Scholar
  7. 7.
    Cook, W.: National travelling salesman problem (2009), http://www.tsp.gatech.edu/index.html
  8. 8.
    Elmaliach, Y., Agmon, N., Kaminka, G.A.: Multi-robot area patrol under frequency constraints. In: IEEE Int. Conf. on Robotics and Automation, Roma, Italy, pp. 385–390 (2007)Google Scholar
  9. 9.
    Gabriely, Y., Rimon, E.: Competitive on-line coverage of grid environments by a mobile robot. Computational Geometry: Theory and Applications 24(3), 197–224 (2003)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hussein, I.I., Stipanovic̀, D.M.: Effective coverage control for mobile sensor networks with guaranteed collision avoidance. IEEE Transactions on Control Systems Technology 15(4), 642–657 (2007)CrossRefGoogle Scholar
  11. 11.
    Lin, S., Kernighan, B.: Effective heuristic algorithm for the traveling salesman problem. Operations Research 21, 498–516 (1973)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Michael, N., Stump, E., Mohta, K.: Persistent surveillance with a team of mavs. In: IEEE/RSJ Int. Conf. on Intelligent Robots & Systems, San Francisco, CA, pp. 2708–2714 (2011)Google Scholar
  13. 13.
    Papadimitriou, C.H., Yannakakis, M.: The traveling salesman problem with distances one and two. Math. Oper. Res. 18, 1–11 (1993)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Pasqualetti, F., Franchi, A., Bullo, F.: On optimal cooperative patrolling. In: IEEE Conf. on Decision and Control, Atlanta, GA, USA, pp. 7153–7158 (2010)Google Scholar
  15. 15.
    Smith, R.N., Schwager, M., Smith, S.L., Rus, D., Sukhatme, G.S.: Persistent ocean monitoring with underwater gliders: Adapting sampling resolution. Journal of Field Robotics 28(5), 714–741 (2011)CrossRefGoogle Scholar
  16. 16.
    Smith, S.L., Rus, D.: Multi-robot monitoring in dynamic environments with guaranteed currency of observations. In: IEEE Conf. on Decision and Control, Atlanta, GA, pp. 514–521 (2010)Google Scholar
  17. 17.
    Tiwari, A., Jun, M., Jeffcoat, D.E., Murray, R.M.: Analysis of dynamic sensor coverage problem using Kalman filters for estimation. In: IFAC World Congress, Prague, Czech Republic (2005)Google Scholar
  18. 18.
    Tulabandhula, T., Rudin, C., Jaillet, P.: Machine learning and the traveling repairman (2011), http://arxiv.org/abs/1104.5061

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Soroush Alamdari
    • 1
  • Elaheh Fata
    • 2
  • Stephen L. Smith
    • 2
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of Electrical and Computer EngineeringUniversity of WaterlooWaterlooCanada

Personalised recommendations