Advertisement

Theory of Computing Systems

, Volume 42, Issue 3, pp 289–305 | Cite as

Grasp and Delivery for Moving Objects on Broken Lines

  • Yuichi AsahiroEmail author
  • Eiji Miyano
  • Shinichi Shimoirisa
Article

Abstract

This paper studies the following variant of the Vehicle Routing Problem that we call the Grasp and Delivery for Moving Objects (GDMO) problem, motivated by robot navigation: The input to the problem consists of n products, each of which moves on a predefined path with a fixed constant speed, and a robot arm of capacity one. In each round, the robot arm grasps one product and then delivers it to the depot. The goal of the problem is to find a collection of tours such that the robot arm grasps and delivers as many products as possible. In this paper we prove the following results: (i) If the products move on broken lines with at least one bend, then the GDMO is MAXSNP-hard, and (ii) it can be approximated with ratio 2. However, (iii) if we impose the “straight line without bend” restriction on the motion of every product, then the GDMO becomes tractable.

Keywords

Moving objects Grasp and delivery Approximation algorithm MAXSNP-hardness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal, P.K., Sharir, M.: Davenport-Schinzel sequences and their geometric applications. In: Sack, J., Urutia, J. (eds.) Handbook of Computational Geometry. Elsevier, Amsterdam (1999) Google Scholar
  2. 2.
    Asahiro, Y., Horiyama, T., Makino, K., Ono, H., Sakuma, T., Yamashita, M.: How to collect balls moving in the Euclidean plane. Discrete Appl. Math. 154(16), 2247–2262 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Asahiro, Y., Miyano, E., Shimoirisa, S.: K-collect tours for moving objects with release times and deadlines. In: Proc. Systemics, Cybernetics and Informatics, vol. III, pp. 192–197, 2005 Google Scholar
  4. 4.
    Bansal, N., Blum, A., Chawla, S., Meyerson, A.: Approximation algorithms for deadline-TSP and vehicle routing with time-windows. In: Proc. ACM Symposium on Theory of Computing, pp. 166–174, 2004 Google Scholar
  5. 5.
    Blum, A., Chawla, S., Karger, D., Lane, T., Meyerson, A., Minkoff, M.: Approximation algorithms for orienteering and discounted-reward TSP. In: Proc. IEEE Symposium on Foundations of Computer Science, pp. 46–55, 2003 Google Scholar
  6. 6.
    Bar-Noy, A., Guha, S., Naor, J., Schieber, B.: Approximating the throughput of multiple machines under real-time scheduling. SIAM J. Comput. 31(2), 331–352 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chalasani, P., Motwani, R., Rao, A.: Approximation algorithms for robot grasp and delivery. In: Proc. International Workshop on Algorithmic Foundations of Robotics, pp. 347–362, 1996 Google Scholar
  8. 8.
    Chuzhoy, J., Ostrovsky, R., Rabani, Y.: Approximation algorithms for the job interval selection problem and related scheduling problem. In: Proc. IEEE Symposium on Foundations of Computer Science, pp. 348–356, 2001 Google Scholar
  9. 9.
    Desrochers, M., Desrosiers, J., Solomon, M.M.: A new optimization algorithm for the vehicle routing problem with time windows. Oper. Res. 40, 342–354 (1992) zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hammar, M., Nilsson, B.J.: Approximation results for kinetic variants of TSP. In: Proc. International Colloquium on Automata, Languages and Programming, pp. 392–401, 1999 Google Scholar
  11. 11.
    Helvig, C.S., Robins, G., Zelikovsky, A.: Moving target TSP and related problems. In: Proc. European Symposium on Algorithms, pp. 453–464, 1998 Google Scholar
  12. 12.
    Karger, D., Stein, C., Wein, J.: Scheduling algorithms. In: Atallah, M.J. (ed.) Handbook of Algorithms and Theory of Computation. CRC Press, Boca Raton (1997) Google Scholar
  13. 13.
    Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.): The Traveling Salesman Problem. Wiley, Chichester (1985) zbMATHGoogle Scholar
  14. 14.
    Lenstra, J.K., Rinnooy Kan, A.H.G., Brucker, P.: Complexity of machine scheduling problems. Ann. Discrete Math. 1, 343–362 (1977) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Moore, J.M.: An n job, one machine sequencing algorithm for minimizing the number of late jobs. Manag. Sci. 15, 102–109 (1968) zbMATHGoogle Scholar
  16. 16.
    Papadimitriou, C.H.: Computational Complexity. Addison–Wesley, Reading (1994) zbMATHGoogle Scholar
  17. 17.
    Spieksma, F.C.R.: On the approximability of an interval scheduling problem. J. Sched. 2, 215–227 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Toth, P., Vigo, D.: An overview of vehicle routing problems. In: Tosh, P., Vigo, D. (eds.) The Vehicle Routing Problem. SIAM, Philadelphia (2001) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Yuichi Asahiro
    • 1
    Email author
  • Eiji Miyano
    • 2
  • Shinichi Shimoirisa
    • 2
  1. 1.Department of Social Information SystemsKyushu Sangyo UniversityFukuokaJapan
  2. 2.Department of Systems Innovation and InformaticsKyushu Institute of TechnologyFukuokaJapan

Personalised recommendations