Maximum Locally Stable Matchings

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 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 μ of hospitals and residents has a local blocking pair (h, r) if (h, r) is a blocking pair of μ, and there is a resident r′ so that r′ is simultaneously an employee of h 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. This paper continues the study of locally stable matchings, focusing on those with maximum cardinality. We refer to them as maximum locally stable matchings. First, we present families of instances where finding a maximum locally stable matchings is computationally easy. For one family of instances, every stable matching is a maximum locally stable matching. This family includes the case when G is a complete graph. For the other family of instances, every maximum cardinality matching is a maximum locally stable matching. This family includes the case when G is an empty graph. Next, we provide a bound on how good a stable matching approximates a maximum locally stable matching based on the size of a maximum matching of Ḡ, the complement of G. An implication of this bound is that when G is almost a complete graph, a stable matching is almost a maximum locally stable matching. We then consider the case when G is almost an empty graph and show that finding a maximum locally stable matchings is still easy. Nonetheless, finding a maximum locally stable matching is in general computationally hard. In particular, we prove that finding a locally stable matching of a certain size is NP-complete and that approximating the size of a maximum locally stable matching within 21/19− δ is NP-hard.