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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

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 [

We emphasize that a locally stable matching can contain blocking pairs, but these blocking pairs are unlikely to compromise the matching. This may seem odd—if

The first part of Arcaute and Vassilvitskii’s paper [

First, 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

Next, we show that when

Finally, in spite of the results above, we show that finding a maximum locally stable matching is computationally hard in general. 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

The rest of the paper is organized as follows. In

In the stable matchings literature, the problem of finding a stable matching in the Arcaute–Vassilvitskii model sans the social network is often referred to as the Hospital/Residents problem (HR). The firms correspond to the hospitals while the workers correspond to the residents. In their seminal paper on stable matchings [

It is not difficult to see that

For HR instance

Let us suppose that all hospitals in

When not all hospitals in

The previous example shows that the bound in Proposition 1 is tight.

Following the above terminology, we shall call the problem of finding a locally stable matching and a maximum locally stable matching in the Arcaute–Vassilvitskii’s model HR+SN and max-HR+SN respectively, where SN stands for

We note though that the converse of Proposition 2 is not always true as shown by this simple instance

In the next proposition, we provide a bound similar to Proposition 1.

When we appended our running example with the social network consisting of a clique containing

Suppose

We have shown that

In

In this section, we present a family of HR+SN instances where finding a maximum locally stable matching can be solved in polynomial time.

So suppose some firms in

The next theorem provides a bound that is different from the one presented in Proposition 3. It shows that when

Now suppose some firms in

We now consider the opposite case of Theorem 1.

In this next theorem, we consider the case when

It is also straightforward to verify that when

To find a maximum locally stable matching of

There are at most

An SMI (Stable Marriage with Incomplete Lists) instance is just like an HR instance, only that all firms have capacity 1. Hence, all of its stable matchings have the same size. An SMTI (Stable Marriage with Ties and Incomplete Lists) instance is an SMI instance except that the participants’ preference lists are allowed to contain ties. For this problem, a pair

Let

For (ii), suppose

What we have shown is that as long as a locally stable matching of

Checking whether

Finally, we note from (i) that a maximum locally stable matching of

In the statement of Theorem 5, we simply described what edges should be in

Consistent ties arise naturally when firms and/or workers derive their preference lists from a

We now translate this result to HR+SN.

Next, we argue that max-HR+SN is NP-hard to approximate by appealing to the details of the following result of Halldórsson

Clearly,

Theorem 7 provides a lower bound while Proposition 3 provides an upper bound to the factor of the best approximation algorithm for max-HR+SN. Can this gap be narrowed? We suspect that the answer is yes since the source of our hardness results, max-SMTI, has a number of

Now, suppose some firms in