Abstract
We consider the following stochastic optimization problem first introduced by Chen et al. in [7]. We are given a vertex set of a random graph where each possible edge is present with probability p e . We do not know which edges are actually present unless we scan/probe an edge. However whenever we probe an edge and find it to be present, we are constrained to picking the edge and both its end points are deleted from the graph. We wish to find the maximum matching in this model. We compare our results against the optimal omniscient algorithm that knows the edges of the graph and present a 0.573 factor algorithm using a novel sampling technique. We also prove that no algorithm can attain a factor better than 0.898 in this model.
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References
Adamczyk, M.: Improved analysis of the greedy algorithm for stochastic matching. Inf. Process. Lett. 111(15), 731–737 (2011)
Aronson, J., Dyer, M., Frieze, A., Suen, S.: Randomized greedy matching. ii. Random Struct. Algorithms 6, 55–73 (1995)
Aronson, J., Frieze, A., Pittel, B.G.: Maximum matchings in sparse random graphs: Karp-sipser revisited. Random Struct. Algorithms 12, 111–177 (1998)
Bansal, N., Gupta, A., Li, J., Mestre, J., Nagarajan, V., Rudra, A.: When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part II. LNCS, vol. 6347, pp. 218–229. Springer, Heidelberg (2010)
Birnbaum, B., Mathieu, C.: On-line bipartite matching made simple. SIGACT News 39, 80–87 (2008)
Chebolu, P., Frieze, A., Melsted, P.: Finding a maximum matching in a sparse random graph in o(n) expected time. J. ACM 57, 24:1–24:27 (2010)
Chen, N., Immorlica, N., Karlin, A.R., Mahdian, M., Rudra, A.: Approximating Matches Made in Heaven. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 266–278. Springer, Heidelberg (2009)
Dean, B.C., Goemans, M.X., Vondrák, J.: Adaptivity and approximation for stochastic packing problems. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 395–404. Society for Industrial and Applied Mathematics, Philadelphia (2005)
Edmonds, J.: Paths, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)
Farkas, J.G.: Uber die theorie der einfachen ungleichungen. Journal fur die Reine und Angewandte Mathematik 124, 1–27 (1902)
Feldman, J., Mehta, A., Mirrokni, V.S., Muthukrishnan, S.: Online stochastic matching: Beating 1-1/e. In: FOCS, pp. 117–126 (2009)
Frieze, A., Pittel, B.: Perfect matchings in random graphs with prescribed minimal degree. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2003, pp. 148–157. Society for Industrial and Applied Mathematics, Philadelphia (2003)
Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: SODA, pp. 982–991 (2008)
Karande, C., Mehta, A., Tripathi, P.: Online bipartite matching in the unknown distributional model. In: STOC, pp. 106–117 (2011)
Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for online bipartite matching. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (1990)
Mahdian, M., Yan, Q.: Online bipartite matching with random arrivals: An approach based on strongly factor-revealing lps. In: STOC, pp. 117–126 (2011)
Manshadi, V.H., Oveis-Gharan, S., Saberi, A.: Online stochastic matching: Online actions based on offline statistics. In: SODA (2011)
Mehta, A., Mirrokni, V.: Online ad serving: Theory and practice. Tutorial (2011)
Nikolova, E., Kelner, J.A., Brand, M., Mitzenmacher, M.: Stochastic Shortest Paths Via Quasi-convex Maximization. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 552–563. Springer, Heidelberg (2006)
Papadimitriou, C.H., Yannakakis, M.: Shortest paths without a map. Theor. Comput. Sci. 84, 127–150 (1991)
Karp, R.M., Sipser, M.: Maximum matching in sparse random graphs. In: FOCS, pp. 364–375 (1981)
Ross, L.F., Rubin, D.T., Siegler, M., Josephson, M.A., Thistlethwaite, J.R., Woodle, E.S.: The case for a living emotionally related international kidney donor exchange registry. Transplantation Proceedings 18, 5–9 (1986)
Ross, L.F., Rubin, D.T., Siegler, M., Josephson, M.A., Thistlethwaite, J.R., Woodle, E.S.: Ethics of a paired-kidney-exchange program. The New England Journal of Medicine 336, 1752–1755 (1997)
Shmoys, D.B., Swamy, C.: An approximation scheme for stochastic linear programming and its application to stochastic integer programs. J. ACM 53, 978–1012 (2006)
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Costello, K.P., Tetali, P., Tripathi, P. (2012). Stochastic Matching with Commitment. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_69
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DOI: https://doi.org/10.1007/978-3-642-31594-7_69
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