Each vertex of a graph initially may contain an object of a known type. A final state, specifying the type of object desired at each vertex, is also given. A single vehicle of unit capacity is available for shipping objects among the vertices. The swapping problem is to compute a shortest route such that a vehicle can accomplish the rearrangement of the objects while following this route. We exhibit several structural properties of shortest routes and develop polynomial approximation algorithms that are variations of a well‐known “patching” algorithm for the traveling salesman problem. We prove tight constant performance guarantees for these algorithms and note as a side product that these bounds hold and are tight also for the latter problem.