Advertisement

A polynomial-time approximation scheme for the airplane refueling problem

  • Iftah Gamzu
  • Danny Segev
Article

Abstract

We consider the airplane refueling problem that was introduced by Gamow and Stern in their classical book Puzzle-Math (1958). In this setting, we wish to deliver a bomb the farthest possible distance, being much greater than the range of any individual airplane at our disposal. For this purpose, the only feasible option is to better utilize our fleet via mid-air refueling. Starting with a fleet of airplanes that can instantaneously refuel one another and gradually drop out of formation, how would we design the best refueling policy, i.e., one that maximizes the distance traveled by the last remaining plane? Even though Gamow and Stern provided an elegant characterization of the optimal refueling policy for the special case of identical airplanes, the general problem with arbitrary tank volumes and consumption rates has remained widely open, as pointed out by Woeginger (Albers et al., Dagstuhl seminar proceedings 10071, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany, 2010). To our knowledge, other than a logarithmic approximation, which can be attributed to folklore, improved performance guarantees have not been obtained to date. In this paper, we propose a polynomial-time approximation scheme for the airplane refueling problem in its utmost generality. Our approach employs widely-known techniques related to geometric rounding, time stretching, guessing arguments, and timeline partitions. These are augmented by additional insight and ideas, that enable us to devise reductions to well-structured instances of generalized assignment and to exploit LP-rounding algorithms for the latter problem. We complement this result by presenting a fast and easy-to-implement algorithm that attains a constant factor approximation for the optimal refueling policy.

Keywords

Scheduling Approximation algorithms PTAS Generalized assignment 

References

  1. Afrati, F. N., Bampis, E., Chekuri, C., Karger, D. R., Kenyon, C., Khanna, S., et al. (1999). Approximation schemes for minimizing average weighted completion time with release dates. In 40th Annual IEEE symposium on foundations of computer science (pp. 32–44).Google Scholar
  2. Albers, S., Baruah, S. K., Möhring, R. H., & Pruhs, K. (2010). 10071 open problems—scheduling. In Dagstuhl seminar proceedings 10071. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany.Google Scholar
  3. Alon, N., & Spencer, J. H. (2016). The probabilistic method (4th ed.). Hoboken: Wiley.Google Scholar
  4. Baltag, A., van Ditmarsch, H., & Moss, L. (2008). Epistemic logic and information update. In P. Adriaans & J. van Benthem (Eds.), Handbook of the philosophy of information (pp. 361–456). Amsterdam: Elsevier.CrossRefGoogle Scholar
  5. Bansal, N., & Pruhs, K. (2014). The geometry of scheduling. SIAM Journal on Computing, 43(5), 1684–1698.CrossRefGoogle Scholar
  6. Chekuri, C., & Khanna, S. (2004). Approximation algorithms for minimizing average weighted completion time. In J. Y. T. Leung (Ed.), Handbook of scheduling: Algorithms, models, and performance analysis. Boca Raton: CRC Press.Google Scholar
  7. Chekuri, C., & Khanna, S. (2005). A polynomial time approximation scheme for the multiple knapsack problem. SIAM Journal on Computing, 35(3), 713–728.CrossRefGoogle Scholar
  8. Cheung, M., Mestre, J., Shmoys, D. B., & Verschae, J. (2017). A primal-dual approximation algorithm for min-sum single-machine scheduling problems. SIAM Journal on Discrete Mathematics, 31(2), 825–838.CrossRefGoogle Scholar
  9. Chuzhoy, J., Ostrovsky, R., & Rabani, Y. (2006). Approximation algorithms for the job interval selection problem and related scheduling problems. Mathematics of Operations Research, 31(4), 730–738.CrossRefGoogle Scholar
  10. Cohen, R., Katzir, L., & Raz, D. (2006). An efficient approximation for the generalized assignment problem. Information Processing Letters, 100(4), 162–166.CrossRefGoogle Scholar
  11. Correa, J. R., Skutella, M., & Verschae, J. (2012). The power of preemption on unrelated machines and applications to scheduling orders. Mathematics of Operations Research, 37(2), 379–398.CrossRefGoogle Scholar
  12. De, S., & Sen, A. (1996). The generalised Gamow–Stern problem. The Mathematical Gazette, 80(488), 345–348.CrossRefGoogle Scholar
  13. Feige, U., & Vondrák, J. (2010). The submodular welfare problem with demand queries. Theory of Computing, 6(1), 247–290.CrossRefGoogle Scholar
  14. Feldman, M., & Naor, J. S. (2017). Non-preemptive buffer management for latency sensitive packets. Journal of Scheduling, 20(4), 337–353.CrossRefGoogle Scholar
  15. Fiat, A., Mansour, Y., & Nadav, U. (2008). Competitive queue management for latency sensitive packets. In Proceedings of the 19th annual ACM-SIAM symposium on discrete algorithms (pp. 228–237).Google Scholar
  16. Fleischer, L., Goemans, M. X., Mirrokni, V. S., & Sviridenko, M. (2011). Tight approximation algorithms for maximum separable assignment problems. Mathematics of Operations Research, 36(3), 416–431.CrossRefGoogle Scholar
  17. Gamow, G., & Stern, M. (1958). Puzzle-math. New York City: Viking Press.Google Scholar
  18. Gandhi, R., Halldórsson, M. M., Kortsarz, G., & Shachnai, H. (2008). Improved bounds for scheduling conflicting jobs with minsum criteria. ACM Transactions on Algorithms, 4(1), 11:1–11:20.CrossRefGoogle Scholar
  19. Hall, L. A., Schulz, A. S., Shmoys, D. B., & Wein, J. (1997). Scheduling to minimize average completion time: Off-line and on-line approximation algorithms. Mathematics of Operations Research, 22(3), 513–544.CrossRefGoogle Scholar
  20. Höhn, W. (2014). Complex single machine scheduling: Theoretical and practical aspects of sequencing. Ph.D. thesis, TU Berlin.Google Scholar
  21. Höhn, W., Mestre, J., & Wiese, A. (2014). How unsplittable-flow-covering helps scheduling with job-dependent cost functions. In Proceedings of the 41st international colloquium on automata, languages, and programming (pp. 625–636).Google Scholar
  22. Knuth, D. E. (1969). The Gamow–Stern elevator problem. Journal of Recreational Mathematics, 2, 131–137.Google Scholar
  23. Lübbecke, E., Maurer, O., Megow, N., & Wiese, A. (2016). A new approach to online scheduling: Approximating the optimal competitive ratio. ACM Transactions on Algorithms, 13(1), 15:1–15:34.CrossRefGoogle Scholar
  24. Megow, N. & Verschae, J. (2013). Dual techniques for scheduling on a machine with varying speed. In Proceedings of the 40th international colloquium on automata, languages, and programming (pp. 745–756).Google Scholar
  25. Nutov, Z., Beniaminy, I., & Yuster, R. (2006). A (1–1/ e)-approximation algorithm for the generalized assignment problem. Operations Research Letters, 34(3), 283–288.CrossRefGoogle Scholar
  26. Procaccia, A. D. (2013). Cake cutting: Not just child’s play. Communications of the ACM, 56(7), 78–87.CrossRefGoogle Scholar
  27. Shmoys, D. B., & Tardos, É. (1993). An approximation algorithm for the generalized assignment problem. Mathematical Programming, 62, 461–474.CrossRefGoogle Scholar
  28. Sitters, R. (2014). Polynomial time approximation schemes for the traveling repairman and other minimum latency problems. In Proceedings of the 25th annual ACM-SIAM symposium on discrete algorithms (pp. 604–616).Google Scholar
  29. Thomson, W. (2011). Fair allocation rules. In K. J. Arrow, A. Sen, & K. Suzumura (Eds.), Handbook of social choice and welfare (Vol. 2, pp. 393–506). Amsterdam: North-Holland Publishing Co.CrossRefGoogle Scholar
  30. van Ditmarsch, H., Ruan, J., & Verbrugge, R. (2008). Sum and product in dynamic epistemic logic. Journal of Logic and Computation, 18(4), 563–588.CrossRefGoogle Scholar
  31. Wuffle, A. (1982). The pure theory of elevators. Mathematics Magazine, 55(1), 30–37.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Amazon ResearchHaifaIsrael
  2. 2.Department of StatisticsUniversity of HaifaHaifaIsrael

Personalised recommendations