Sorting of Permutations by Cost-Constrained Transpositions

TL;DR

This paper develops polynomial-time algorithms for decomposing permutations into transpositions with non-uniform costs, providing exact solutions for metric-path costs and approximate solutions for general costs, with applications across multiple fields.

Contribution

It introduces new polynomial-time algorithms for permutation decomposition under cost constraints, including exact methods for metric-path costs and approximation algorithms for general costs.

Findings

Polynomial-time algorithms for non-uniform transposition costs

Exact solutions for metric-path cost functions

Approximate algorithms for arbitrary cost functions

Abstract

We address the problem of finding the minimum decomposition of a permutation in terms of transpositions with non-uniform cost. For arbitrary non-negative cost functions, we describe polynomial-time, constant-approximation decomposition algorithms. For metric-path costs, we describe exact polynomial-time decomposition algorithms. Our algorithms represent a combination of Viterbi-type algorithms and graph-search techniques for minimizing the cost of individual transpositions, and dynamic programing algorithms for finding minimum cost cycle decompositions. The presented algorithms have applications in information theory, bioinformatics, and algebra.