Let $\pi$ be an arbitrary permutation of the set $\lbrace 1,\ldots,n,n+1,\ldots,2n \rbrace$ for some $n \in \mathbb{N}$. We call a swap local if you swap two neighboring positions in $\pi$, i.e. if you change the positions $i$ and $i-1$ or $i$ and $i+1$ for some $i$.
A $c$-separation of the pairs $(1,n+1),\ldots,(n,2n)$ is a partition of $\pi$ in $L := \lbrace \pi(1), \ldots, \pi(p) \rbrace$ and $R := \lbrace \pi(p+1), \ldots, \pi(2n) \rbrace$ such that for at least $c$ pairs $(k,k+n)$ hold $(k,k+n) \in L \times R$ or $(k,k+n) \in R \times L$.
What is a good upper bound on the number of local swaps I have to perform on $\pi$ to get a $c$-separation of the pairs $(1,n+1), \ldots, (n,2n)$?