# 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}
}
• A. Caprara
• Published 1999
• Computer Science
• SIAM J. Discret. Math.
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
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 Science
INFORMS 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
- #«# 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
• A. Caprara
• Computer Science, Mathematics
RECOMB
• 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
• A. Caprara
• Mathematics, Computer Science
J. 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
• A. Labarre
• Computer Science, Mathematics
SIAM 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, Mathematics
J. 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
• Mathematics
ArXiv
• 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 Science
SODA '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
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.

## References

SHOWING 1-10 OF 18 REFERENCES
On the Tightness of the Alternating-Cycle Lower Bound for Sorting by Reversals
• A. Caprara
• Mathematics, Computer Science
J. 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.
A column-generation based branch-and-bound algorithm for sorting by reversals
• Computer Science
Mathematical Support for Molecular Biology
• 1998
An exact branch-and-bound algorithm for SBR is proposed, which can solve to optimality SBR instances of considerably larger size with respect to previous existing methods.
A 3/2-approximation algorithm for sorting by reversals
A polynomial-time 3/2-approximation algorithm for the problem of determining the smallest number of reversals required to transform a given permutation to the identity (or to sort the permutation by reversals).
To cut…or not to cut (applications of comparative physical maps in molecular evolution)
• Mathematics
SODA '96
• 1996
The strong Kecedoglu-Sankoff conjecture is proved: for every permutation there exists an optimal sorting by reversals that never increases the number of breakpoints and a new algorithm is presented based on the notion of the spin of a permutation.
Fast Sorting by Reversal
• Computer Science, Mathematics
CPM
• 1996
This paper exploits a few combinatorial properties of the cycle graph of a permutation and proposes an O(n2α(n) implementation of the algorithm where α is the inverse Ackerman function and improves implementations of the other rearrangement distance problems.
A compendium of NP optimization problems
• Computer Science, Mathematics
WWW Spring 1994
• 1994
This compendium of approximability results of NP-hard optimization problems has been collected together and is interested in studying a class of optimization problems whose feasible solutions are short and easy-to-recognize.
Genome rearrangements and sorting by reversals
• Computer Science
Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science
• 1993
It is demonstrated that the expected reversal distance is very close to the reversal diameter thereby indicating that reversal distance provides a good separation between related and non-related sequences.
The NP-Completeness of Some Edge-Partition Problems
We show that for each fixed $n \geqq 3$ it is NP-complete to determine whether an arbitrary graph can be edge-partitioned into subgraphs isomorphic to the complete graph $K_n$. The NP-completeness
Faster and simpler algorithm for sorting signed permutations by reversals
• Computer Science
SODA '97
• 1997
A quadratic algorithm for finding the minimum number of reversals needed to sort a signed permutation and considerably simplifies the combinatorial structures used by the analysis.
Efficient Bounds for Oriented Chromosome Inversion Distance
• Computer Science
CPM
• 1994
It is shown that tight bounds on the minimum number of reversals can be found by simple and efficient algorithms.