# Sorting Permutations by Reversals and Eulerian Cycle Decompositions

@article{Caprara1999SortingPB, title={Sorting Permutations by Reversals and Eulerian Cycle Decompositions}, author={Alberto Caprara}, journal={SIAM J. Discret. Math.}, year={1999}, volume={12}, pages={91-110} }

We analyze the strong relationship among three combinatorial problems, namely, the problem of sorting a permutation by the minimum number of reversals (MIN-SBR), the problem of finding the maximum number of edge-disjoint alternating cycles in a breakpoint graph associated with a given permutation (MAX-ACD), and the problem of partitioning the edge set of an Eulerian graph into the maximum number of cycles (MAX-ECD). We first illustrate a nice characterization of breakpoint graphs, which leads…

## 204 Citations

SORTING PERMUTATIONS BY REVERSALS AND EULERIAN CYCLE

- Computer Science
- 1999

A transformation from MAX-ACD to MIN-SBR is described, which is therefore shown to be NP-hard as well, answering an outstanding question which has been open for some years.

Sorting Permutations by Reversals Through Branch-and-Price

- Computer ScienceINFORMS J. Comput.
- 2001

We describe an exact algorithm for the problem of sorting a permutation by the minimum number of reversals, originating from evolutionary studies in molecular biology. Our approach is based on an…

Sorting Permutations by Reversals Through Branch-and-Price

- Computer Science
- 2001

- #«# e describe an exact algorithm for the problem of sorting a permutation by the minimum number of reversals, originating from evolutionary studies in molecular biology. Our approach is based on…

Formulations and hardness of multiple sorting by reversals

- Computer Science, MathematicsRECOMB
- 1999

This work describes a graph-theoretic relaxation of MSBR, which is the counterpart of the so-called alternating-cycle decomposition relaxation for SBR, and uses this relaxation to show that, even if the number of given permutations equals 3, MSBR is NP-hard, and hence so is nee SBR.

On the Tightness of the Alternating-Cycle Lower Bound for Sorting by Reversals

- Mathematics, Computer ScienceJ. Comb. Optim.
- 1999

The problem of sorting by reversals and its alternating-cycle relaxation are essentially the same problem, with the exception of a small fraction of “pathological” instances, justifying the use of algorithms which are heavily based on this relaxation.

Lower Bounding Edit Distances between Permutations

- Computer Science, MathematicsSIAM J. Discret. Math.
- 2013

This paper presents an algebraic reinterpretation of the cycle graph of a permutation $pi$ as an even permutation $\overline{\pi}$ and shows how to reformulate the authors' sorting problems in terms of particular factorizations of the latter permutation.

Improved Approximation for Breakpoint Graph Decomposition and Sorting by Reversals

- Computer Science, MathematicsJ. Comb. Optim.
- 2002

This paper shows how to improve the approximation ratio achievable in polynomial time for BGD, from the previously known 3-2, by using the best known approximation algorithms for Stable Set on graphs with maximum degree 4 as well as for Set Packing where the maximum size of a set is 6.

Algorithms for the Maximum Eulerian Cycle Decomposition Problem

- MathematicsArXiv
- 2022

Given an Eulerian graph G, in the Maximum Eulerian Cycle Decomposition problem, we are interested in finding a collection of edge-disjoint cycles {E1, E2, . . . , Ek} in G such that all edges of G…

Fast practical solution of sorting by reversals

- Computer ScienceSODA '00
- 2000

The results show that, although the problem is hard and the exact algorithm proposed has apparently exponential running time even on average, the instances of practical interest can be solved to proven optimality very fast.

Edit Distances and Factorisations of Even Permutations

- Computer Science, MathematicsESA
- 2008

This paper presents an algebraic reinterpretation of the cycle graph of the permutation to sort as an even permutation, and obtains a new lower bound on the prefix transposition distance, which is shown to outperform previous results.

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