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

TL;DR

This paper proves that finding a perfect bipartite matching with the fewest intersecting edges in a planar embedded bipartite graph is NP-complete, highlighting computational complexity in geometric graph problems.

Contribution

It establishes the NP-completeness of minimizing intersecting edges in perfect matchings within planar bipartite graphs, a special case of token swapping problems.

Findings

Problem is NP-complete

Equivalent to a special case of token swapping

Highlights complexity in geometric graph optimization

Abstract

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.