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 N
P-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 $bn(M) le 2sqrt{10} cdot bn(M_{rm NC})$, where $bn(M)$ is the length of a longest edge in $M$. An interesting implication of our construction is that $bn(M_{rm NC})/bn(M_{rm C}) le 2sqrt{10}$.
