, Volume 80, Issue 5, pp 1556–1574 | Cite as

Towards Flexible Demands in Online Leasing Problems

  • Shouwei Li
  • Christine Markarian
  • Friedhelm Meyer auf der Heide
Part of the following topical collections:
  1. Special Issue on Computing and Combinatorics


We consider online leasing problems in which demands arrive over time and need to be served by leasing resources. We introduce a new model for these problems in which a resource can be leased for K different durations each incurring a different cost (longer leases cost less per time unit). Each demand i can be served any time between its arrival \(a_i\) and its deadline \(a_i + d_i\) by a leased resource. The objective is to meet all deadlines while minimizing the total leasing costs. This model is a natural generalization of Meyerson’s ParkingPermitProblem (in: Proceedings of the 46th annual IEEE symposium on foundations of computer science, FOCS ’05, IEEE Computer Society, Washington, pp 274–284, 2005) in which \(d_i=0\) for all i. We propose an online algorithm that is \(\varTheta (K + \frac{d_\textit{max}}{l_\textit{min}})\)-competitive, where \(d_\textit{max}\) and \(l_\textit{min}\) denote the largest \(d_i\) and the shortest available lease length, respectively. We also extend SetCoverLeasing and FacilityLeasing to their respective variants in which deadlines are added. For the former, we give an \(\mathcal {O}\left( \log (m \cdot (K + \frac{d_\textit{max}}{l_\textit{min}}))\log l_\textit{max} \right) \)-competitive randomized algorithm, where m represents the number of subsets and \(l_\textit{max}\) represents the largest available lease length. This improves on existing solutions for the original SetCoverLeasing problem. For the latter, we give an \(\mathcal {O}\left( (K + \frac{d_\textit{max}}{l_\textit{min}})\log l_{\text {max}} \right) \)-competitive deterministic algorithm.


Online algorithms Leasing Infrastructure problems Parking permit problem Deadlines Set cover leasing Facility leasing 


  1. 1.
    Abshoff, S., Kling, P., Markarian, C., Meyer auf der Heide, F., Pietrzyk, P.: Towards the price of leasing online. J. Comb. Optim. 32(4), 1197–1216 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abshoff, S., Markarian, C., Meyer auf der Heide, F.: Randomized online algorithms for set cover leasing problems. In: Combinatorial Optimization and Applications—8th International Conference, COCOA 2014, Wailea, Maui, HI, USA, 19–21 December 2014, Proceedings, pp. 25–34 (2014)Google Scholar
  3. 3.
    Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: The online set cover problem. SIAM J. Comput. 39(2), 361–370 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.S.: A general approach to online network optimization problems. ACM Trans. Algorithms 2(4), 640–660 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Alon, N., Azar, Y., Gutner, S.: Admission control to minimize rejections and online set cover with repetitions. In: Proceedings of the Seventeenth Annual ACM Symposium on Parallelism in Algorithms and Architectures, SPAA ’05, pp. 238–244. ACM, New York (2005)Google Scholar
  6. 6.
    Alon, N., Moshkovitz, D., Safra, S.: Algorithmic construction of sets for k-restrictions. ACM Trans. Algorithms 2(2), 153–177 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Anthony, B.M., Gupta, A.: Infrastructure leasing problems. In: IPCO, pp. 424–438 (2007)Google Scholar
  8. 8.
    Ausiello, G., Giannakos, A., Paschos, V.T.: Greedy algorithms for online set covering and related problems. In: Proceedings of the Twelfth Computing: The Australasian Theory Symposium—vol. 51, CATS ’06, pp. 145–151. Australian Computer Society, Inc., Darlinghurst (2006)Google Scholar
  9. 9.
    Awerbuch, B., Azar, Y.: Buy-at-bulk network design. In: 38th Annual Symposium on Foundations of Computer Science, FOCS ’97, Miami Beach, Florida, USA, 19–22 October 1997, pp. 542–547 (1997)Google Scholar
  10. 10.
    Berman, P., DasGupta, B.: Approximating the online set multicover problems via randomized winnowing. Theor. Comput. Sci. 393(1–3), 54–71 (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    Berman, P., DasGupta, B., Sontag, E.D.: Randomized approximation algorithms for set multicover problems with applications to reverse engineering of protein and gene networks. Discrete Appl. Math. 155(6–7), 733–749 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bhawalkar, K., Gollapudi, S., Panigrahi, D.: Online set cover with set requests. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, 4–6 September 2014, Barcelona, Spain, pp. 64–79 (2014)Google Scholar
  13. 13.
    Buchbinder, N., Naor, J.: Online primal–dual algorithms for covering and packing. Math. Oper. Res. 34(2), 270–286 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chudak, F.A., Williamson, D.P.: Improved approximation algorithms for capacitated facility location problems. Math. Program. 102(2), 207–222 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chvatal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fotakis, D.: On the competitive ratio for online facility location. In: Proceedings of the 30th International Conference on Automata, Languages and Programming, ICALP’03, pp. 637–652. Springer, Berlin (2003)Google Scholar
  18. 18.
    Fotakis, D.: A primal–dual algorithm for online non-uniform facility location. J. Discrete Algorithms 5(1), 141–148 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Freund, A., Rawitz, D.: Combinatorial interpretations of dual fitting and primal fitting. In: Approximation and Online Algorithms, First International Workshop, WAOA 2003, Budapest, Hungary, 16–18 September 2003, Revised Papers, pp. 137–150 (2003)Google Scholar
  20. 20.
    Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’98, pp. 649–657. Society for Industrial and Applied Mathematics, Philadelphia (1998)Google Scholar
  21. 21.
    Gupta, A., Kumar, A., Pál, M., Roughgarden, T.: Approximation via cost sharing: simpler and better approximation algorithms for network design. J. ACM 54(3), 11 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J. ACM 50(6), 795–824 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-median problems using the primal–dual schema and Lagrangian relaxation. J. ACM 48(2), 274–296 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9(3), 256–278 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kling, P., Meyer auf der Heide, F., Pietrzyk, P.: An algorithm for online facility leasing. In: Structural Information and Communication Complexity—19th International Colloquium, SIROCCO 2012, Reykjavik, Iceland, June 30–July 2, 2012, Revised Selected Papers, pp. 61–72 (2012)Google Scholar
  26. 26.
    Korman, S.: On the use of randomization in the online set cover problem. Master’s thesis, Weizmann Institute of Science, Rehovot (2004)Google Scholar
  27. 27.
    Li, S.: A 1.488 approximation algorithm for the uncapacitated facility location problem. In: Proceedings of the 38th International Conference on Automata, Languages and Programming—Volume Part II, ICALP’11, pp. 77–88. Springer, Berlin (2011)Google Scholar
  28. 28.
    Li, S., Mäcker, A., Markarian, C., Meyer auf der Heide, F., Riechers, S.: Towards flexible demands in online leasing problems. In: Computing and Combinatorics—21st International Conference, COCOON 2015, Beijing, China, 4–6 August 2015, Proceedings, pp. 277–288 (2015)Google Scholar
  29. 29.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13(4), 383–390 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mahdian, M., Ye, Y., Zhang, J.: Improved approximation algorithms for metric facility location problems. In: Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization, pp. 229–242 (2002)Google Scholar
  31. 31.
    Mettu, R.R., Plaxton, C.G.: The online median problem. SIAM J. Comput. 32(3), 816–832 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Meyerson, A.: Online facility location. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, FOCS ’01, pp. 426–431. IEEE Computer Society, Washington (2001)Google Scholar
  33. 33.
    Meyerson, A.M.: The parking permit problem. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’05, pp. 274–284. IEEE Computer Society, Washington (2005)Google Scholar
  34. 34.
    Nagarajan, C., Williamson, D.P.: Offline and online facility leasing. Discrete Optim. 10(4), 361–370 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Salman, F.S., Cheriyan, J., Ravi, R., Subramanian, S.: Buy-at-bulk network design: approximating the single-sink edge installation problem. In: Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’97, pp. 619–628. Society for Industrial and Applied Mathematics, Philadelphia (1997)Google Scholar
  36. 36.
    Shmoys, D.B., Tardos, E., Aardal, K.: Approximation algorithms for facility location problems (extended abstract). In: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, STOC ’97, pp. 265–274. ACM, New York (1997)Google Scholar
  37. 37.
    Vazirani, V.V.: Approximation Algorithms. Springer, New York (2001)zbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Computer Science Department, Heinz Nixdorf InstituteUniversity of PaderbornPaderbornGermany

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