Complexity of Finding Perfect Bipartite Matchings Minimizing the Number of Intersecting Edges

Consider a problem where we are given a bipartite graph H with vertices arranged on two horizontal lines in the plane, such that the two sets of vertices placed on the two lines form a bipartition of H. We additionally require that H admits a perfect matching and assume that edges of H are embedded in the plane as segments. The goal is to compute the minimal number of intersecting edges in a perfect matching in H. The problem stems from so-called token swapping problems, introduced by Yamanaka et al. [3] and generalized by Bonnet, Miltzow and Rzazewski [1]. We show that our problem, equivalent to one of the special cases of one of the token swapping problems, is NP-complete.