An Optimal Online Algorithm for Weighted Bipartite Matching and Extensions to Combinatorial Auctions

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8125)


We study online variants of weighted bipartite matching on graphs and hypergraphs. In our model for online matching, the vertices on the right-hand side of a bipartite graph are given in advance and the vertices on the left-hand side arrive online in random order. Whenever a vertex arrives, its adjacent edges with the corresponding weights are revealed and the online algorithm has to decide which of these edges should be included in the matching. The studied matching problems have applications, e.g., in online ad auctions and combinatorial auctions where the right-hand side vertices correspond to items and the left-hand side vertices to bidders.

Our main contribution is an optimal algorithm for the weighted matching problem on bipartite graphs. The algorithm is a natural generalization of the classical algorithm for the secretary problem achieving a competitive ratio of e ≈ 2.72 which matches the well-known upper and lower bound for the secretary problem. This shows that the classic algorithmic approach for the secretary problem can be extended from the simple selection of a best possible singleton to a rich combinatorial optimization problem.

On hypergraphs with (d + 1)-uniform hyperedges, corresponding to combinatorial auctions with bundles of size d, we achieve competitive ratio O(d) in comparison to the previously known ratios \(O\big(d^2\big)\) and O(d logm), where m is the number of items. Additionally, we study variations of the hypergraph matching problem representing combinatorial auctions for items with bounded multiplicities or for bidders with submodular valuation functions. In particular for the case of submodular valuation functions we improve the competitive ratio from O(logm) to e.


Bipartite Graph Competitive Ratio Online Algorithm Combinatorial Auction Submodular Function 
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.


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  1. 1.
    Aggarwal, G., Goel, G., Karande, C., Mehta, A.: Online vertex-weighted bipartite matching and single-bid budgeted allocations. In: SODA, pp. 1253–1264 (2011)Google Scholar
  2. 2.
    Azar, Y., Regev, O.: Combinatorial algorithms for the unsplittable flow problem. Algorithmica 44(1), 49–66 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Babaioff, M., Immorlica, N., Kleinberg, R.D.: Matroids, secretary problems, and online mechanisms. In: SODA, pp. 434–443 (2007)Google Scholar
  4. 4.
    Birnbaum, B.E., Mathieu, C.: On-line bipartite matching made simple. SIGACT News 39(1), 80–87 (2008)CrossRefGoogle Scholar
  5. 5.
    Buchbinder, N., Jain, K., Singh, M.: Secretary problems via linear programming. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 163–176. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Chakraborty, S., Lachish, O.: Improved competitive ratio for the matroid secretary problem. In: SODA, pp. 1702–1712 (2012)Google Scholar
  7. 7.
    Devanur, N.R., Hayes, T.P.: The adwords problem: online keyword matching with budgeted bidders under random permutations. In: ACM Conference on Electronic Commerce, pp. 71–78 (2009)Google Scholar
  8. 8.
    Devanur, N.R., Jain, K., Kleinberg, R.D.: Randomized primal-dual analysis of ranking for online bipartite matching. In: SODA, pp. 101–107 (2013)Google Scholar
  9. 9.
    Dimitrov, N.B., Plaxton, C.G.: Competitive weighted matching in transversal matroids. Algorithmica 62(1-2), 333–348 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Feldman, M., Naor, J., Schwartz, R.: Improved competitive ratios for submodular secretary problems (extended abstract). In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) RANDOM/APPROX 2011. LNCS, vol. 6845, pp. 218–229. Springer, Heidelberg (2011)Google Scholar
  11. 11.
    Ferguson, T.S.: Who solved the secretary problem? Statistical Science, 282–289 (1989)Google Scholar
  12. 12.
    Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: SODA, pp. 982–991 (2008)Google Scholar
  13. 13.
    Im, S., Wang, Y.: Secretary problems: Laminar matroid and interval scheduling. In: SODA, pp. 1265–1274 (2011)Google Scholar
  14. 14.
    Jaillet, P., Soto, J.A., Zenklusen, R.: Advances on matroid secretary problems: Free order model and laminar case. CoRR abs/1207.1333 (2012)Google Scholar
  15. 15.
    Kalyanasundaram, B., Pruhs, K.: Online weighted matching. J. Algorithms 14(3), 478–488 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Karande, C., Mehta, A., Tripathi, P.: Online bipartite matching with unknown distributions. In: STOC, pp. 587–596 (2011)Google Scholar
  17. 17.
    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
  18. 18.
    Kleinberg, R.D.: A multiple-choice secretary algorithm with applications to online auctions. In: SODA, pp. 630–631 (2005)Google Scholar
  19. 19.
    Korula, N., Pál, M.: Algorithms for secretary problems on graphs and hypergraphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part II. LNCS, vol. 5556, pp. 508–520. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Krysta, P., Vöcking, B.: Online mechanism design (randomized rounding on the fly). In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 636–647. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  21. 21.
    Mahdian, M., Yan, Q.: Online bipartite matching with random arrivals: an approach based on strongly factor-revealing lps. In: STOC, pp. 597–606 (2011)Google Scholar
  22. 22.
    Mehta, A., Saberi, A., Vazirani, U.V., Vazirani, V.V.: Adwords and generalized online matching. J. ACM 54(5) (2007)Google Scholar
  23. 23.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic game theory. Cambridge University Press (2007)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceCornell UniversityIthacaUSA
  2. 2.Department of Computer ScienceRWTH Aachen UniversityGermany

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