Bottleneck Non-Crossing Matching in the Plane

TL;DR

This paper investigates the computational complexity of finding bottleneck non-crossing matchings in the plane, proving NP-hardness, and provides an approximation algorithm with a specific performance guarantee.

Contribution

The paper establishes NP-hardness of the problem, shows it cannot be approximated within any PTAS, and introduces an efficient algorithm with a bounded approximation ratio.

Findings

Proves the problem is NP-hard.

Shows no PTAS exists for the problem.

Provides an $O(n^{1.5}\log^{0.5} n)$-time approximation algorithm.

Abstract

Let $P$ be a set of $2n$ points in the plane, and let $M_{\rm C}$ (resp., $M_{\rm NC}$) denote a bottleneck matching (resp., a bottleneck non-crossing matching) of $P$. We study the problem of computing $M_{\rm NC}$. We first prove that the problem is NP-hard and does not admit a PTAS. Then, we present an $O(n^{1.5}\log^{0.5} n)$-time algorithm that computes a non-crossing matching $M$ of $P$, such that $bn(M) \le 2\sqrt{10} \cdot bn(M_{\rm NC})$, where $bn(M)$ is the length of a longest edge in $M$. An interesting implication of our construction is that $bn(M_{\rm NC})/bn(M_{\rm C}) \le 2\sqrt{10}$.