1 research outputs found
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.Comment: 14 pages, 4 figure