, Volume 6, Issue 2, pp 157-182

Improved Approximation for Breakpoint Graph Decomposition and Sorting by Reversals

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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 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.