Abstract
Consider a random graph model where each possible edge e is present independently with some probability p e . We are given these numbers p e , and want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, we are forced to add it to our matching. Further, each vertex i is allowed to be queried at most t i times. How should we adaptively query the edges to maximize the expected weight of the matching? We consider several matching problems in this general framework (some of which arise in kidney exchanges and online dating, and others arise in modeling online advertisements); we give LP-rounding based constant-factor approximation algorithms for these problems. Our main results are:
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We give a 5.75-approximation for weighted stochastic matching on general graphs, and a 5-approximation on bipartite graphs. This answers an open question from [Chen et al. ICALP 09].
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Combining our LP-rounding algorithm with the natural greedy algorithm, we give an improved 3.88-approximation for unweighted stochastic matching on general graphs and 3.51-approximation on bipartite graphs.
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We introduce a generalization of the stochastic online matching problem [Feldman et al. FOCS 09] that also models preference-uncertainty and timeouts of buyers, and give a constant factor approximation algorithm.
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References
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. arXiv (2010)
Bhattacharya, S., Goel, G., Gollapudi, S., Munagala, K.: Budget constrained auctions with heterogeneous items. In: STOC (2009), arxiv:abs/0907.4166
Birnbaum, B.E., Mathieu, C.: On-line bipartite matching made simple. SIGACT News 39(1), 80–87 (2008)
Chen, N., Immorlica, N., Karlin, A.R., Mahdian, M., Rudra, A.: Approximating matches made in heaven. In: ICALP, Part I, vol. (1), pp. 266–278 (2009)
Dean, B.C., Goemans, M.X., Vondrák, J.: Adaptivity and approximation for stochastic packing problems. In: SODA, pp. 395–404 (2005)
Dean, B.C., Goemans, M.X., Vondrák, J.: Approximating the stochastic knapsack problem: the benefit of adaptivity. Math. Oper. Res. 33(4), 945–964 (2008), http://dx.doi.org/10.1287/moor.1080.0330
Feldman, J., Mehta, A., Mirrokni, V.S., Muthukrishnan, S.: Online stochastic matching: Beating 1 − 1/e. In: FOCS (2009), Arxiv:abs/0905.4100
Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: SODA, pp. 982–991 (2008)
Guha, S., Munagala, K.: Approximation algorithms for partial-information based stochastic control with markovian rewards. In: FOCS, pp. 483–493 (2007)
Guha, S., Munagala, K.: Multi-armed bandits with metric switching costs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 496–507. Springer, Heidelberg (2009)
Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted sampling: approximation algorithms for stochastic optimization. In: STOC, pp. 417–426. ACM, New York (2004)
Kalyanasundaram, B., Pruhs, K.: Online weighted matching. J. Algorithms 14(3), 478–488 (1993)
Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: STOC, pp. 352–358 (1990)
Katriel, I., Kenyon-Mathieu, C., Upfal, E.: Commitment under uncertainty: Two-stage stochastic matching problems. Theoretical Computer Science 408(2-3), 213–223 (2008)
Mahdian, M., Nazerzadeh, H., Saberi, A.: Allocating online advertisement space with unreliable estimates. In: EC. p. 294 (2007)
Mehta, A., Saberi, A., Vazirani, U.V., Vazirani, V.V.: Adwords and generalized on-line matching. In: FOCS, pp. 264–273 (2005)
Schrijver, A.: Combinatorial Optimization. Springer, Heidelberg (2003)
Shmoys, D.B., Tardos, É.: An approximation algorithm for the generalized assignment problem. Math. Program. 62, 461–474 (1993)
Shmoys, D.B., Swamy, C.: An approximation scheme for stochastic linear programming and its application to stochastic integer programs. J. ACM 53(6), 1012 (2006)
Swamy, C., Shmoys, D.B.: Approximation algorithms for 2-stage stochastic optimization problems. ACM SIGACT News 37(1), 46 (2006)
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Bansal, N., Gupta, A., Li, J., Mestre, J., Nagarajan, V., Rudra, A. (2010). When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6347. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15781-3_19
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