Abstract
Motivated by the observation that most companies are more likely to consider job applicants referred by their employees than those who applied on their own, Arcaute and Vassilvitskii modeled a job market that integrates social networks into stable matchings in an interesting way. We call their model HR+SN because an instance of their model is an ordered pair (I, G) where I is a typical instance of the Hospital/Residents problem (HR) and G is a graph that describes the social network (SN) of the residents in I. A matching p, of hospitals and residents has a local blocking pair (h, r) if (h, r) is a blocking pair of ii, and there is a resident r' such that r' is simultaneously an employee of h in the matching and a neighbor of r in G. Such a pair is likely to compromise the matching because the participants have access to each other through r': r can give her resume to r' who can then forward it to h. A locally stable matching is a matching with no local blocking pairs. The cardinality of the locally stable matchings of I can vary. This paper presents a variety of results on computing a locally stable matching with maximum cardinality.
Highlights
Motivated by the observation that most companies are more likely to consider job applicants suggested by their employees than those who apply on their own, Arcaute and Vassilvitskii [1] modeled a job market that integrates social networks into stable matchings
We show that when Ḡ, the complement of G, has a maximum matching of size r, the size of a maximum locally stable matching of the instance is at most r more than the size of a stable matching of the instance
We show that when G has a constant number of edges—i.e., G is almost an empty graph—finding a maximum locally stable matching can still be done in polynomial time
Summary
Motivated by the observation that most companies are more likely to consider job applicants suggested by their employees than those who apply on their own, Arcaute and Vassilvitskii [1] modeled a job market that integrates social networks into stable matchings. The second part of their paper examines the evolution of the job market They consider a decentralized version of Gale and Shapley’s algorithm and show that for a specific case the algorithm converges to a locally stable matching under weak stochastic conditions. We present families of instances where the problem of finding a maximum locally stable matchings is computationally easy. For one family of instances, every stable matching of the instance is a maximum locally stable matching This family includes the case when G, the social network of the workers, is a complete graph. For the other family of instances, every maximum matching of the firms and workers is a maximum locally stable matching This family includes the case when G is an empty graph.
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