Decomposing Permutations via Cost-Constrained Transpositions

We consider the problem of finding the minimum cost transposition decomposition of a permutation. In this framework, arbitrary non-negative costs are assigned to individual transpositions and the task at hand is to devise polynomial-time, constant-approximation decomposition algorithms. We describe a polynomial-time algorithm based on specialized search strategies that constructs the optimal decomposition of individual transpositions. The analysis of the optimality of decompositions of single transpositions uses graphical models and Menger's theorem. We also present a dynamic programing algorithms that finds the minimum cost, minimum length decomposition of a cycle and show that this decomposition represents a 4-approximation of the optimal solution. The results presented for individual cycles extend to general permutations.