Discrete Optimization A factor 1 2 approximation algorithm for two-stage stochastic matching problems Nan Kong, Andrew J. Schaefer * Department of Industrial Engineering, University of Pittsburgh, 1048 Benedum Hall, Pittsburgh, PA 15261, USA Received 26 November 2003; accepted 22 October 2004 Abstract We introduce the two-stage stochastic maximum-weight matching problem and demonstrate that this problem is NP-complete. We give a factor 1 2 approximation algorithm and prove its correctness. We also provide a tight example to show the bound given by the algorithm is exactly 1 2 . Computational results on some two-stage stochastic bipartite matching instances indicate that the performance of the approximation algorithm appears to be substantially better than its worst-case performance. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Stochastic programming; Approximation algorithms; Matching; Combinatorial optimization 1. Introduction Let G =(V, E) be a graph, and let each edge e 2 E have an edge weight c e . The maximum- weight matching problem (Cook et al., 1998) is max X e2E c e x e X e2dðvÞ x e 6 1;      8v 2 V ; x e 2f0; 1g; 8 e 2 E ( ) : ð1Þ It is well known that the maximum-weight match- ing problem is polynomially solvable (Edmonds, 1965). Consider a stochastic programming exten- sion of this problem as follows. Each edge has two weights, a first-stage weight c e , and a discretely distributed second-stage weight ~ d e . The first-stage decision x is to choose a matching in G. After the decision, a scenario of the second-stage edge weights is realized. That is, each edge weight is as- signed to one of the r possible values d 1 e ; ... ; d r e with corresponding probabilities p 1 , ..., p r . For each scenario s = 1, ..., r, the second-stage deci- sion y s is to choose a matching over those vertices unmatched by the first-stage matching. Without loss of generality, the edge weights c e and d s e for 0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.10.011 * Corresponding author. Tel.: +1 412 624 5045; fax: +1 412 624 9831. E-mail address: schaefer@ie.pitt.edu (A.J. Schaefer). European Journal of Operational Research xxx (2004) xxx–xxx www.elsevier.com/locate/dsw ARTICLE IN PRESS