Tight Exact and Approximate Algorithmic Results on Token Swapping

Given a graph $G=(V,E)$ with $V=\{1,\ldots,n\}$, we place on every vertex a token $T_1,\ldots,T_n$. A swap is an exchange of tokens on adjacent vertices. We consider the algorithmic question of finding a shortest sequence of swaps such that token $T_i$ is on vertex $i$. We are able to achieve essentially matching upper and lower bounds, for exact algorithms and approximation algorithms. For exact algorithms, we rule out $2^{o(n)}$ algorithm under ETH. This is matched with a simple $2^{O(n\log n)}$ algorithm based on dynamic programming. We give a general $4$-approximation algorithm and show APX-hardness. Thus, there is a small constant $\delta>1$ such that every polynomial time approximation algorithm has approximation factor at least $\delta$. Our results also hold for a generalized version, where tokens and vertices are colored. In this generalized version each token must go to a vertex with the same color.