Advertisement

Block Sorting Is APX-Hard

  • N. S. Narayanaswamy
  • Swapnoneel Roy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9079)

Abstract

Block Sorting is an NP-hard combinatorial optimization problem motivated by applications in Computational Biology and Optical Character Recognition (OCR). It has been approximated in P time within a factor of 2 using two different techniques and the complexity of better approximations has been open for close to a decade now. In this work we prove that Block Sorting does not admit a PTAS unless P = NP i.e. it is APX-Hard. The hardness result is based on new properties, that we identify, of the existing NP-hardness reduction from E3-SAT to Block Sorting. In an attempt to obtain an improved approximation for Block Sorting, we consider a generalization of the well-studied Block Merging, called \(k\)-Block Merging which is defined for each \(k \ge 1\), and the \(1\)- Block Merging problem is the same as the Block Merging problem. We show that the optimum \(k\)-Block Merging is an \(1+ \frac{1}{k}\)-approximation to the optimum block sorting. We then show that for each \(k \ge 2\), we prove \(k\)-Block Merging to be NP-Hard, thus proving a dichotomy result associated with block sorting.

Keywords

Approximation Ratio Truth Assignment Block Move Blue Edge Blue Component 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bafna, V., Pevzner, P.A.: Sorting by Transpositions. SIAM Journal of Discrete Mathematics 11(2), 224–240 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bein, W.W., Larmore, L.L., Latifi, S., Sudborough, I.H.: A Quadratic TIme 2-Approximation Algorithm for Block Sorting. Theoretical Computer Science 410(8–10), 711–717 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bein, W.W., Larmore, L.L., Latifi, S., Sudborough, I.H.: Block sorting is hard. International Journal of Foundations of Computer Science 14(3), 425–437 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bulteau, L., Fertin, G., Rusu, I.: Sorting by Transpositions is Difficult. Automata, Languages and Programming 6755, 654–665 (2011)MathSciNetGoogle Scholar
  5. 5.
    Christie, D.A.: Genome Rearrangement Problems. PhD Thesis, University of Glasgow (1999)Google Scholar
  6. 6.
    Elias, I., Hartman, T.: A 1.375-Approximation Algorithm for Sorting by Transpositions. IEEE/ACM Transactions on Computational Biology and Bioinformatics 3(4), 369–379 (2006). doi: 10.1109/TCBB.2006.44 CrossRefGoogle Scholar
  7. 7.
    Gobi, R., Latifi, S., Bein, W.W.: Adaptive Sorting Algorithms for Evaluation of Automatic Zoning Employed in OCR Devices. In: Proceedings of the 2000 International Conference on Imaging Science, Systems, and Technology, CISST 2000, pp. 253–259. CSREA Press (2000)Google Scholar
  8. 8.
    Håstad, J.: Some Optimal Inapproximability Results. Journal of the ACM 48(4), 798–859 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Mahajan, M., Rama, R., Raman, V., Vijayakumar, S.: Merging and Sorting ByStrip Moves. In: Pandya, P.K., Radhakrishnan, J. (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 314–325. Springer, Heidelberg (2003)Google Scholar
  10. 10.
    Mahajan, M., Rama, R., Raman, V., Vijayakumar, S.: lApproximate Block Sorting. International Journal of Foundation of Computer Science, 337–356 (2006)Google Scholar
  11. 11.
    Mahajan, M., Rama, R., Vijayakumar, S.: Block sorting: a characterization and some heuristics. Nordic Journal of Computing 14(1), 126–150 (2007)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Mahajan, M., Rama, R., Vijayakumar, S.: Towards constructing optimal block move sequences. In: Chwa, K.-Y., Munro, J.I. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 33–42. Springer, Heidelberg (2004)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology MadrasChennaiIndia
  2. 2.School of ComputingUniversity of North FloridaJacksonvilleUSA

Personalised recommendations