We address the problem of finding the minimum decomposition of a permutation in terms of transpositions with non-uniform cost. For metric-path costs, we describe exact polynomial-time decomposition algorithms. For extended-metric-path cost functions, we describe polynomial-time constant-approximation decomposition algorithms. Our algorithms rely on graphical representations of permutations and graph-search techniques for minimizing the permutation decomposition cost. The presented algorithms have applications in information theory, bioinformatics, and algebra.