Abstract
We study the online stochastic matching problem in a form motivated by Internet display advertisement. Recently, Feldman et al. gave an algorithm that achieves 0.6702 competitive ratio, thus breaking through the 1 − 1/e barrier. One of the questions left open in their work is to obtain a better competitive ratio by generalizing their two suggested matchings (TSM) algorithm to d-suggested matchings (d-SM).
We show that the best competitive ratio that can be obtained with the static analysis used in the d-SM algorithm is upper bounded by 0.76, even for the special case of d-regular graphs, thus suggesting that a dynamic analysis may be needed to improve the competitive ratio significantly. We make the first step in this direction by showing that the RANDOM algorithm, which assigns an impression to a randomly chosen eligible advertiser, achieves \(1-e^{-d}d^d/d!=1-O(1/\sqrt{d})\) competitive ratio for d-regular graphs, which converges to 1 as d increases. On the hardness side, we improve the upper bound of 0.989 on the competitive ratio of any online algorithm obtained by Feldman et al. to 1 − 1/(e + e 2) ≈ 0.902. Finally, we show how to modify the TSM algorithm to obtain an improved 0.699 approximation for general bipartite graphs.
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References
Karp, R., Vazirani, U., Vazirani, V.: An optimal algorithm for online bipartite matching. In: STOC (1990)
Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: SODA (2008)
Feldman, J., Mehta, A., Mirrokni, V., Muthukrishnan, S.: Online stochastic matching: Beating 1 − 1/e. In: FOCS (2009)
Azar, Y., Birnbaum, B., Karlin, A., Mathieu, C., Nguyen, C.: Improved approximation algorithms for budgeted allocations. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 186–197. Springer, Heidelberg (2008)
Srinivasan, A.: Budgeted allocations in the full-information setting. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX and RANDOM 2008. LNCS, vol. 5171, pp. 247–253. Springer, Heidelberg (2008)
Mehta, A., Saberi, A., Vazirani, U., Vazirani, V.: Adwords and generalized online matching. In: FOCS (2005)
Buchbinder, N., Jain, K., Naor, J.: Online primal-dual algorithms for maximizing ad-auctions. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 253–264. Springer, Heidelberg (2007)
Kalyanasundaram, B., Pruhs, K.R.: An optimal deterministic algorithm for online b -matching. Theoretical Computer Science (2000)
Devanur, N., Hayes, T.: The adwords problem: online keyword matching with budgeted bidders under random permutations. In: EC (2009)
Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)
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Bahmani, B., Kapralov, M. (2010). Improved Bounds for Online Stochastic Matching. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15775-2_15
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DOI: https://doi.org/10.1007/978-3-642-15775-2_15
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