Skip to main content
SpringerLink
Log in

Grasp and Delivery for Moving Objects on Broken Lines

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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

  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

  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

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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

  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

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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

  11. Helvig, C.S., Robins, G., Zelikovsky, A.: Moving target TSP and related problems. In: Proc. European Symposium on Algorithms, pp. 453–464, 1998

  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. Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.): The Traveling Salesman Problem. Wiley, Chichester (1985)

    MATH  Google Scholar 

  14. Lenstra, J.K., Rinnooy Kan, A.H.G., Brucker, P.: Complexity of machine scheduling problems. Ann. Discrete Math. 1, 343–362 (1977)

    Article  MathSciNet  Google Scholar 

  15. Moore, J.M.: An n job, one machine sequencing algorithm for minimizing the number of late jobs. Manag. Sci. 15, 102–109 (1968)

    MATH  Google Scholar 

  16. Papadimitriou, C.H.: Computational Complexity. Addison–Wesley, Reading (1994)

    MATH  Google Scholar 

  17. Spieksma, F.C.R.: On the approximability of an interval scheduling problem. J. Sched. 2, 215–227 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. Department of Social Information Systems, Kyushu Sangyo University, Fukuoka, 813-8503, Japan

    Yuichi Asahiro

  2. Department of Systems Innovation and Informatics, Kyushu Institute of Technology, Fukuoka, 820-8502, Japan

    Eiji Miyano & Shinichi Shimoirisa

Authors
  1. Yuichi Asahiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eiji Miyano
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shinichi Shimoirisa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuichi Asahiro.

Additional information

Y. Asahiro’s research was supported in part by Grant-in-Aid for Young Scientists 15700021, and Research on Priority Areas 16092223.

E. Miyano’s research was supported in part by Grant-in-Aid for Research on Priority Areas 16092223.

Presently, S. Shimoirisa is with the Software Development Engineering Department, Products Development Center, Retail Information Systems Company, Toshiba TEC Corporation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Asahiro, Y., Miyano, E. & Shimoirisa, S. Grasp and Delivery for Moving Objects on Broken Lines. Theory Comput Syst 42, 289–305 (2008). https://doi.org/10.1007/s00224-007-9081-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-007-9081-y

Keywords

Search

Navigation