TY - JOUR

T1 - When LP is the cure for your matching woes

T2 - Improved bounds for stochastic matchings

AU - Bansal, Nikhil

AU - Gupta, Anupam

AU - Li, Jian

AU - Mestre, Julián

AU - Nagarajan, Viswanath

AU - Rudra, Atri

N1 - Publisher Copyright:
© Springer Science+Business Media, LLC 2011.

PY - 2012/8/1

Y1 - 2012/8/1

N2 - Consider a random graph model where each possible edge e is present independently with some probability pe. 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 ti 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 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 preference-uncertainty and timeouts of buyers, and give a constant factor approximation algorithm.

AB - Consider a random graph model where each possible edge e is present independently with some probability pe. 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 ti 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 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 preference-uncertainty and timeouts of buyers, and give a constant factor approximation algorithm.

KW - Dependent rounding

KW - Online dating

KW - Stochastic optimization

KW - Stochastic packing

UR - http://www.scopus.com/inward/record.url?scp=79953712047&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953712047&partnerID=8YFLogxK

U2 - 10.1007/s00453-011-9511-8

DO - 10.1007/s00453-011-9511-8

M3 - Article

AN - SCOPUS:79953712047

SN - 0178-4617

VL - 63

SP - 733

EP - 762

JO - Algorithmica

JF - Algorithmica

IS - 4

ER -