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Improved Approximation for Breakpoint Graph Decomposition and Sorting by Reversals

Author

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  • Alberto Caprara

    (University of Bologna)

  • Romeo Rizzi

    (University of Aarhus, Ny Munkegade)

Abstract

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 suitably-defined 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.

Suggested Citation

  • Alberto Caprara & Romeo Rizzi, 2002. "Improved Approximation for Breakpoint Graph Decomposition and Sorting by Reversals," Journal of Combinatorial Optimization, Springer, vol. 6(2), pages 157-182, June.
    • Handle: RePEc:spr:jcomop:v:6:y:2002:i:2:d:10.1023_a:1013851611274
    DOI: 10.1023/A:1013851611274
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    References listed on IDEAS

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    1. Alberto Caprara, 1999. "On the Tightness of the Alternating-Cycle Lower Bound for Sorting by Reversals," Journal of Combinatorial Optimization, Springer, vol. 3(2), pages 149-182, July.
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    Cited by:

    1. Xin Chen, 2013. "On sorting unsigned permutations by double-cut-and-joins," Journal of Combinatorial Optimization, Springer, vol. 25(3), pages 339-351, April.
    2. Guohui Lin & Tao Jiang, 2004. "A Further Improved Approximation Algorithm for Breakpoint Graph Decomposition," Journal of Combinatorial Optimization, Springer, vol. 8(2), pages 183-194, June.

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    1. Guohui Lin & Tao Jiang, 2004. "A Further Improved Approximation Algorithm for Breakpoint Graph Decomposition," Journal of Combinatorial Optimization, Springer, vol. 8(2), pages 183-194, June.

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