Generalized Stochastic Matching

In this paper, we generalize the recently studied Stochastic Matching problem to more accurately model a significant medical process, kidney exchange, and several other applications. Up until now the Stochastic Matching problem that has been studied was as follows: given a graph G = (V, E), each edge is included in the realized sub-graph of G mutually independently with probability p_e, and the goal is to find a degree-bounded sub-graph Q of G that has an expected maximum matching that approximates the expected maximum matching of the realized sub-graph. This model does not account for possibilities of vertex dropouts, which can be found in several applications, e.g. in kidney exchange when donors or patients opt out of the exchange process as well as in online freelancing and online dating when online profiles are found to be faked. Thus, we will study a more generalized model of Stochastic Matching in which vertices and edges are both realized independently with some probabilities p_v, p_e, respectively, which more accurately fits important applications than the previously studied model. We will discuss the first algorithms and analysis for this generalization of the Stochastic Matching model and prove that they achieve good approximation ratios. In particular, we show that the approximation factor of a natural algorithm for this problem is at least $0.6568$ in unweighted graphs, and $1/2 + ε$ in weighted graphs for some constant $ε> 0$. We further improve our result for unweighted graphs to $2/3$ using edge degree constrained subgraphs (EDCS).