On the Cost of Interchange Rearrangement in Strings

Consider the following optimization problem: given two strings over the same alphabet, transform one to another by a succession of interchanges of two elements. In each interchange the two participating elements exchange positions. An interchange is given a weight that depends on the distance in the string between the two exchanged elements. The object is to minimize the total weight of the interchanges. This problem is a generalization of a classical problem on permutations (where every element appears once). The generalization considers general strings with possibly repeating elements, and a function assigning weights to the interchanges. The generalization to general strings (with unit weights) has been mentioned by Cayley in the 19th century, and its complexity has been an open question since. We solve this open problem, and consider various weight functions