Approximation and Hardness of Token Swapping

Given a graph G = (V,E) with V = {1, . . . , n}, we place on every vertex a token T1, . . . , Tn. 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 Ti 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 any 2o(n) algorithm under the ETH. This is matched with a simple 2O(n logn) algorithm based on a breadth-first search in an auxiliary graph. We show one general 4-approximation and show APX-hardness. Thus, there is a small constant δ > 1 such that every polynomial time approximation algorithm has approximation factor at least δ. 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. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems