# Stable Noncrossing Matchings

## TL;DR

This paper introduces the concept of weakly stable noncrossing matchings in a geometric setting, proving their existence for any instance and providing an efficient algorithm to find such matchings.

## Contribution

It defines weakly stable noncrossing matchings, proves their existence for all cases, and develops an $O(n^2)$ algorithm to compute one.

## Key findings

- Existence of weakly stable noncrossing matchings for any instance.
- Development of an $O(n^2)$ algorithm to find such matchings.
- Formal proof of the existence of these matchings.

## Abstract

Given a set of $n$ men represented by $n$ points lying on a line, and $n$ women represented by $n$ points lying on another parallel line, with each person having a list that ranks some people of opposite gender as his/her acceptable partners in strict order of preference. In this problem, we want to match people of opposite genders to satisfy people's preferences as well as making the edges not crossing one another geometrically. A noncrossing blocking pair w.r.t. a matching $M$ is a pair $(m,w)$ of a man and a woman such that they are not matched with each other but prefer each other to their own partners in $M$, and the segment $(m,w)$ does not cross any edge in $M$. A weakly stable noncrossing matching (WSNM) is a noncrossing matching that does not admit any noncrossing blocking pair. In this paper, we prove the existence of a WSNM in any instance by developing an $O(n^2)$ algorithm to find one in a given instance.