Advertisement

Bicriteria Online Matching: Maximizing Weight and Cardinality

  • Nitish Korula
  • Vahab S. Mirrokni
  • Morteza Zadimoghaddam
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8289)

Abstract

Inspired by online ad allocation problems, many results have been developed for online matching problems. Most of the previous work deals with a single objective, but, in practice, there is a need to optimize multiple objectives. Here, as an illustrative example motivated by display ads allocation, we study a bi-objective online matching problem.

In particular, we consider a set of fixed nodes (ads) with capacity constraints, and a set of online items (pageviews) arriving one by one. Upon arrival of an online item i, a set of eligible fixed neighbors (ads) for the item is revealed, together with a weight w ia for eligible neighbor a. The problem is to assign each item to an eligible neighbor online, while respecting the capacity constraints; the goal is to maximize both the total weight of the matching and the cardinality. In this paper, we present both approximation algorithms and hardness results for this problem.

An (α, β)-approximation for this problem is a matching with weight at least α fraction of the maximum weighted matching, and cardinality at least β fraction of maximum cardinality matching. We present a parametrized approximation algorithm that allows a smooth tradeoff curve between the two objectives: when the capacities of fixed nodes are large, we give a p(1 − 1/e 1/p ), (1 − p)(1 − 1/e 1/1 − p )-approximation for any 0 ≤ p ≤ 1, and prove a ‘hardness curve’ combining several inapproximability results. These upper and lower bounds are always close (with a maximum gap of 9%), and exactly coincide at the point (0.43, 0.43). For small capacities, we present a smooth parametrized approximation curve for the problem between (0,1 − 1/e) and (1/2,0) passing through a (1/3,0.3698)-approximation.

Keywords

Capacity Constraint Competitive Ratio Online Algorithm Hardness Result Parametrized Approximation Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agrawal, S., Wang, Z., Ye, Y.: A dynamic near-optimal algorithm for online linear programming. Working paper posted at http://www.stanford.edu/~yyye/
  2. 2.
    Bhalgat, A., Feldman, J., Mirrokni, V.S.: Online ad allocation with smooth delivery. In: ACM Conference on Knowledge Discovery, KDD (2012)Google Scholar
  3. 3.
    Bilò, V., Flammini, M., Moscardelli, L.: Pareto approximations for the bicriteria scheduling problem. Journal of Parallel and Distributed Computing 66(3), 393–402 (2006)CrossRefzbMATHGoogle Scholar
  4. 4.
    Birnbaum, B., Mathieu, C.: On-line bipartite matching made simple. SIGACT News 39(1), 80–87 (2008)CrossRefGoogle Scholar
  5. 5.
    Buchbinder, N., Jain, K., Naor, J.(S.): Online Primal-Dual Algorithms for Maximizing Ad-Auctions Revenue. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 253–264. Springer, Heidelberg (2007)Google Scholar
  6. 6.
    Buchbinder, N., Naor, J.: Fair online load balancing. In: Proceedings of the Eighteenth Annual ACM Symposium on Parallelism in Algorithms and Architectures, pp. 291–298. ACM (2006)Google Scholar
  7. 7.
    Devanur, N., Hayes, T.: The adwords problem: Online keyword matching with budgeted bidders under random permutations. In: ACM EC (2009)Google Scholar
  8. 8.
    Devanur, N.R., Jain, K., Sivan, B., Wilkens, C.A.: Near optimal online algorithms and fast approximation algorithms for resource allocation problems. In: ACM Conference on Electronic Commerce, pp. 29–38 (2011)Google Scholar
  9. 9.
    Devanur, N.R., Sivan, B., Azar, Y.: Asymptotically optimal algorithm for stochastic adwords. In: ACM Conference on Electronic Commerce, pp. 388–404 (2012)Google Scholar
  10. 10.
    Feldman, J., Henzinger, M., Korula, N., Mirrokni, V.S., Stein, C.: Online stochastic packing applied to display ad allocation. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 182–194. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Feldman, J., Korula, N., Mirrokni, V., Muthukrishnan, S., Pál, M.: Online ad assignment with free disposal. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 374–385. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Feldman, J., Mehta, A., Mirrokni, V., Muthukrishnan, S.: Online stochastic matching: Beating 1 - 1/e. In: FOCS (2009)Google Scholar
  13. 13.
    Flammini, M., Nicosia, G.: On multicriteria online problems. In: Paterson, M. (ed.) ESA 2000. LNCS, vol. 1879, pp. 191–201. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  14. 14.
    Goel, A., Meyerson, A., Plotkin, S.: Approximate majorization and fair online load balancing. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 384–390. Society for Industrial and Applied Mathematics (2001)Google Scholar
  15. 15.
    Goel, A., Meyerson, A., Plotkin, S.: Combining fairness with throughput: Online routing with multiple objectives. Journal of Computer and System Sciences 63(1), 62–79 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Haeupler, B., Mirrokni, V.S., Zadimoghaddam, M.: Online stochastic weighted matching: Improved approximation algorithms. In: Chen, N., Elkind, E., Koutsoupias, E. (eds.) WINE 2011. LNCS, vol. 7090, pp. 170–181. Springer, Heidelberg (2011)Google Scholar
  17. 17.
    Karande, C., Mehta, A., Tripathi, P.: Online bipartite matching with unknown distributions. In: STOC (2011)Google Scholar
  18. 18.
    Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: STOC, pp. 352–358 (1990)Google Scholar
  19. 19.
    Korula, N., Mirrokni, V.S., Yan, Q.: Whole-page ad allocation and. In: Ad Auctions Workshop (2012)Google Scholar
  20. 20.
    Mahdian, M., Yan, Q.: Online bipartite matching with random arrivals: A strongly factor revealing lp approach. In: STOC (2011)Google Scholar
  21. 21.
    Mehta, A., Saberi, A., Vazirani, U.V., Vazirani, V.V.: Adwords and generalized online matching. J. ACM 54(5) (2007)Google Scholar
  22. 22.
    Menshadi, H., OveisGharan, S., Saberi, A.: Offline optimization for online stochastic matching. In: SODA (2011)Google Scholar
  23. 23.
    Mirrokni, V., Gharan, S.O., ZadiMoghaddam, M.: Simultaneous approximations for adversarial and stochastic online budgeted allocation problems. In: SODA (2012)Google Scholar
  24. 24.
    Vee, E., Vassilvitskii, S., Shanmugasundaram, J.: Optimal online assignment with forecasts. In: ACM EC (2010)Google Scholar
  25. 25.
    Wang, C.-M., Huang, X.-W., Hsu, C.-C.: Bi-objective optimization: An online algorithm for job assignment. In: Abdennadher, N., Petcu, D. (eds.) GPC 2009. LNCS, vol. 5529, pp. 223–234. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Nitish Korula
    • 1
  • Vahab S. Mirrokni
    • 1
  • Morteza Zadimoghaddam
    • 2
  1. 1.Google ResearchNew YorkUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations