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Scheduling Transfers of Resources over Time: Towards Car-Sharing with Flexible Drop-Offs

  • Kateřina Böhmová
  • Yann Disser
  • Matúš Mihalák
  • Rastislav Šrámek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)

Abstract

We consider an offline car-sharing assignment problem with flexible drop-offs, in which n users (customers) present their driving demands, and the system aims to assign the cars, initially located at given locations, to maximize the number of satisfied users. Each driving demand specifies the pick-up location and the drop-off location, as well as the time interval in which the car will be used. If a user requests several driving demands, then she is satisfied only if all her demands are fulfilled. We show that minimizing the number of vehicles that are needed to fulfill all demands is solvable in polynomial time. If every user has exactly one demand, we show that for given number of cars at locations, maximizing the number of satisfied users is also solvable in polynomial time. We then study the problem with two locations A and B, and where every user has two demands: one demand for transfer from A to B, and one demand for transfer from B to A, not necessarily in this order. We show that maximizing the number of satisfied users is NP-hard, and even APX-hard, even if all the transfers take exactly the same (non-zero) time. On the other hand, if all the transfers are instantaneous, the problem is again solvable in polynomial time.

Keywords

Interval scheduling Complexity Algorithms Transfer Resources 

Notes

Acknowledgements

The authors wish to thank Peter Widmayer for many useful discussions and helpful comments, as well as Andreas Bärtschi, Barbara Geissmann, Sandro Montanari, Tobias Pröger, and Thomas Tschager for their ideas during early stage discussions on the topic. Kateřina Böhmová is supported by a Google Europe Fellowship in Optimization Algorithms. The project has been partially supported by the Swiss National Science Foundation (SNF) under the grant number 200021_156620.

References

  1. 1.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer Science & Business Media, Heidelberg (2012)zbMATHGoogle Scholar
  2. 2.
    Bar-Yehuda, R., Halldórsson, M.M., Naor, J., Shachnai, H., Shapira, I.: Scheduling split intervals. SIAM J. Comput. 36(1), 1–15 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blin, G., Fertin, G., Vialette, S.: New results for the 2-interval pattern problem. In: Sahinalp, S.C., Muthukrishnan, S.M., Dogrusoz, U. (eds.) CPM 2004. LNCS, vol. 3109, pp. 311–322. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Böhmová, K., Disser, Y., Mihalák, M., Widmayer, P.: Interval selection with machine-dependent intervals. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 170–181. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, vol. 3. MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  6. 6.
    Crescenzi, P.: A short guide to approximation preserving reductions. In: 12th IEEE Conference on Computational Complexity, pp. 262–273. IEEE (1997)Google Scholar
  7. 7.
    Crochemore, M., Hermelin, D., Landau, G.M., Vialette, S.: Approximating the 2-interval pattern problem. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 426–437. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Gabow, H.N., Maheshwari, S.N., Osterweil, L.J.: On two problems in the generation of program test paths. IEEE Trans. Softw. Eng. 2(3), 227–231 (1976)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ghouila-Houri, A.: Caracterisation des matrices totalement unimodulaires. CR Acad. Sci. Paris 254, 1192–1194 (1962)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kolen, A.W.J., Lenstra, J.K., Papadimitriou, C.H., Spieksma, F.C.R.: Interval scheduling: a survey. Naval Res. Logistics (NRL) 54(5), 530–543 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kovalyov, M.Y., Ng, C., Cheng, T.E.: Fixed interval scheduling: models, applications, computational complexity and algorithms. Eur. J. Oper. Res. 178(2), 331–342 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons, Chichester (1998)zbMATHGoogle Scholar
  13. 13.
    Song, Y., Yu, M.: On finding the longest antisymmetric path in directed acyclic graphs. Inf. Process. Lett. 115(2), 377–381 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Kateřina Böhmová
    • 1
  • Yann Disser
    • 2
  • Matúš Mihalák
    • 1
    • 3
  • Rastislav Šrámek
    • 4
  1. 1.Department of Computer ScienceETH ZürichZürichSwitzerland
  2. 2.Department of MathematicsTU BerlinBerlinGermany
  3. 3.Department of Knowledge EngineeringMaastricht UniversityMaastrichtThe Netherlands
  4. 4.Google ZürichZürichSwitzerland

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