Two-sided popular matchings in bipartite graphs with forbidden/forced elements and weights

Two-sided popular matchings in bipartite graphs are a well-known generalization of stable matchings in the marriage setting, and they are especially relevant when preference lists are incomplete. In this case, the cardinality of a stable matching can be as small as half the size of a maximum matching. Popular matchings allow for assignments of larger size while still guaranteeing a certain fairness condition. In fact, stable matchings are popular matchings of minimum size, and a maximum size popular matching can be as large as twice the size of a(ny) stable matching in a given instance. The structure of popular matchings seems to be more complex, and currently less understood, than that of stable matchings. In this paper, we focus on three optimization problems related to popular matchings. First, we give a granular analysis of the complexity of popular matching with forbidden and forced elements problems, thus complementing results from [Cseh and Kavitha, 2016]. In particular, we show that deciding whether there exists a popular matching with (or without) two given edges is NP-Hard. This implies that finding a popular matching of maximum (resp. minimum) weight is NP-Hard and, even if all weights are nonnegative, inapproximable up to a factor 1/2 (resp. up to any factor). A decomposition theorem from [Cseh and Kavitha, 2016] can be employed to give a 1/2 approximation to the maximum weighted popular matching problem with nonnegative weights, thus completely settling the complexity of those problems.