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The Approximability of the Exemplar Breakpoint Distance Problem

  • Zhixiang Chen
  • Bin Fu
  • Binhai Zhu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4041)

Abstract

In this paper we present the first set of approximation and inapproximability results for the Exemplar Breakpoint Distance Problem. Our inapproximability results hold for the simplest case between only two genomes \({\cal G}\) and \({\cal H}\), each containing only one sequence of genes (possibly with repetitions).

– For the general Exemplar Breakpoint Distance Problem, we prove that the problem does not admit any approximation unless P=NP; in fact, this result holds even when a gene appears in \({\cal G}\) (\({\cal H}\)) at most three times.

– Even on a weaker definition of approximation (which we call weak approximation), we show that the problem does not admit a weak approximation with a factor m 1 − − ε, where m is the maximum length of \({\cal G}\) and \({\cal H}\).

– We present a factor-2(1 + logn) approximation for an interesting special case, namely, one of the two genomes is a k-span genome (i.e., all genes in the same gene family are within a distance k = O(logn)), where n is the number of gene families in \({\cal G}\) and \({\cal H}\).

Keywords

Approximation Algorithm Vertex Cover Dynamic Programming Algorithm Conjunctive Normal Form Weak Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bafna, V., Pevzner, P.: Sorting by reversals: Genome rearrangements in plant organelles and evolutionary history of X chromosome. Mol. Bio. Evol. 12, 239–246 (1995)Google Scholar
  2. 2.
    Bryant, D.: The complexity of calculating exemplar distances. In: Sankoff, D., Nadeau, J. (eds.) Comparative Genomics: Empirical and Analytical Approaches to Gene Order Dynamics, Map Alignment, and the Evolution of Gene Families, pp. 207–212. Kluwer Acad. Pub., Dordrecht (2000)Google Scholar
  3. 3.
    Blin, G., Rizzi, R.: Conserved interval distance computation between non-trivial genomes. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 22–31. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Bergeron, A., Stoye, J.: On the similarity of sets of permutations and its applications to genome comparison. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 68–79. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Bereg, S., Zhu, B.: RNA multiple structural alignment with longest common subsequences. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 32–41. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)MATHGoogle Scholar
  7. 7.
    Chvátal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4, 233–235 (1979)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cormen, T., Leiserson, C., Rivest, R.: Introduction to Algorithms. The MIT Press, Cambridge (1990)MATHGoogle Scholar
  9. 9.
    Dur, I., Safra, S.: The importance of being biased. In: Proc. 34th ACM Symp. on Theory Comput. (STOC 2002), pp. 33–42 (2002)Google Scholar
  10. 10.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, San Francisco (1979)MATHGoogle Scholar
  11. 11.
    Hannenhalli, S., Pevzner, P.: Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations by reversals. J. ACM 46(1), 1–27 (1999)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gascuel, O. (ed.): Mathematics of Evolution and Phylogeny. Oxford University Press, Oxford (2004)Google Scholar
  13. 13.
    Johnson, D.: Approximation algorithms for combinatorial problems. J. Comput. System Sci. 9, 256–278 (1974)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Marron, M., Swenson, K., Moret, B.: Genomic distances under deletions and insertions. Theoretical Computer Science 325(3), 347–360 (2004)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Makaroff, C., Palmer, J.: Mitochondrial DNA rearrangements and transcriptional alternatives in the male sterile cytoplasm of Ogura radish. Mol. Cell. Biol. 8, 1474–1480 (1988)Google Scholar
  17. 17.
    Nguyen, C.T., Tay, Y.C., Zhang, L.: Divide-and-conquer approach for the exemplar breakpoint distance. Bioinformatics 21(10), 2171–2176 (2005)CrossRefGoogle Scholar
  18. 18.
    Palmer, J., Herbon, L.: Plant mitochondrial DNA evolves rapidly in structure, but slowly in sequence. J. Mol. Evolut. 27, 87–97 (1988)CrossRefGoogle Scholar
  19. 19.
    Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and sub-constant error-probability PCP characterization of NP. In: Proc. 29th ACM Symp. on Theory Comput. (STOC 1997), pp. 475–484 (1997)Google Scholar
  20. 20.
    Sankoff, D.: Genome rearrangement with gene families. Bioinformatics 16(11), 909–917 (1999)CrossRefGoogle Scholar
  21. 21.
    Sturtevant, A., Dobzhansky, T.: Inversions in the third chromosome of wild races of drosophila pseudoobscura, and their use in the study of the history of the species. Proc. Nat. Acad. Sci. USA 22, 448–450 (1936)CrossRefGoogle Scholar
  22. 22.
    Tannier, E., Sagot, M.-F.: Sorting by reversals in subquadratic time. In: Sahinalp, S.C., Muthukrishnan, S.M., Dogrusoz, U. (eds.) CPM 2004. LNCS, vol. 3109, pp. 1–13. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  23. 23.
    Watterson, G., Ewens, W., Hall, T., Morgan, A.: The chromosome inversion problem. J. Theoretical Biology 99, 1–7 (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Zhixiang Chen
    • 1
  • Bin Fu
    • 2
  • Binhai Zhu
    • 3
  1. 1.Department of Computer ScienceUniversity of Texas-Pan AmericanEdinburgUSA
  2. 2.Department of Computer Science, Research Institute for ChildrenUniversity of New OrleansNew OrleansUSA
  3. 3.Department of Computer ScienceMontana State UniversityBozemanUSA

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