When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings
 Nikhil Bansal,
 Anupam Gupta,
 Jian Li,
 Julián Mestre,
 Viswanath Nagarajan,
 Atri Rudra
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Consider a random graph model where each possible edge e is present independently with some probability p _{ e }. Given these probabilities, we 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 LProunding based constantfactor approximation algorithms for these problems. Our main results are the following:

We give a 4 approximation for weighted stochastic matching on general graphs, and a 3 approximation on bipartite graphs. This answers an open question from Chen et al. (ICALP’09, LNCS, vol. 5555, pp. 266–278, [2009]).

We introduce a generalization of the stochastic online matching problem (Feldman et al. in FOCS’09, pp. 117–126, [2009]) that also models preferenceuncertainty and timeouts of buyers, and give a constant factor approximation algorithm.
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 Title
 When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings
 Journal

Algorithmica
Volume 63, Issue 4 , pp 733762
 Cover Date
 20120801
 DOI
 10.1007/s0045301195118
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Stochastic optimization
 Stochastic packing
 Online dating
 Dependent rounding
 Industry Sectors
 Authors

 Nikhil Bansal ^{(1)}
 Anupam Gupta ^{(2)}
 Jian Li ^{(3)}
 Julián Mestre ^{(4)}
 Viswanath Nagarajan ^{(1)}
 Atri Rudra ^{(5)}
 Author Affiliations

 1. IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA
 2. Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, 15213, USA
 3. Computer Science Department, University of Maryland, College Park, MD, USA
 4. School of Information Technologies, University of Sydney, Sydney, NSW, Australia
 5. Department of Computer Science and Engineering, University at Buffalo, SUNY, Buffalo, NY, 14260, USA