Improved Approximation for Breakpoint Graph Decomposition and Sorting by Reversals
 Alberto Caprara,
 Romeo Rizzi
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Sorting by Reversals (SBR) is one of the most widely studied models of genome rearrangements in computational molecular biology. At present, \(\frac{3}{2}\) is the best known approximation ratio achievable in polynomial time for SBR. A very closely related problem, called Breakpoint Graph Decomposition (BGD), calls for a largest collection of edge disjoint cycles in a suitablydefined graph. It has been shown that for almost all instances SBR is equivalent to BGD, in the sense that any solution of the latter corresponds to a solution of the former having the same value. In this paper, we show how to improve the approximation ratio achievable in polynomial time for BGD, from the previously known \(\frac{3}{2}\) to \(\frac{{33}}{{23}} + \varepsilon \) for any ε > 0. Combined with the results in (Caprara, Journal of Combinatorial Optimization, vol. 3, pp. 149–182, 1999b), this yields the same approximation guarantee for n! − O((n − 5)!) out of the n! instances of SBR on permutations with n elements. Our result uses 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. Any improvement in the ratio achieved by these approximation algorithms will yield an automatic improvement of our result.
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 Title
 Improved Approximation for Breakpoint Graph Decomposition and Sorting by Reversals
 Journal

Journal of Combinatorial Optimization
Volume 6, Issue 2 , pp 157182
 Cover Date
 20020601
 DOI
 10.1023/A:1013851611274
 Print ISSN
 13826905
 Online ISSN
 15732886
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 sorting by reversals
 breakpoint graph
 alternating cycle decomposition
 set packing
 stable set
 approximation algorithm
 Industry Sectors
 Authors

 Alberto Caprara ^{(1)}
 Romeo Rizzi ^{(2)}
 Author Affiliations

 1. DEIS, University of Bologna, Viale Risorgimento 2, I40136, Bologna, Italy
 2. BRICS, Department of Computer Science, University of Aarhus, Ny Munkegade, DK8000, Aarhus C, Denmark