Document Open Access Logo

Approximation Guarantees for Shortest Superstrings: Simpler and Better

Authors Matthias Englert , Nicolaos Matsakis , Pavel Veselý



PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2023.29.pdf
  • Filesize: 0.78 MB
  • 17 pages

Document Identifiers

Author Details

Matthias Englert
  • University of Warwick, Coventry, UK
Nicolaos Matsakis
  • Charles University, Prague, Czech Republic
Pavel Veselý
  • Charles University, Prague, Czech Republic

Cite AsGet BibTex

Matthias Englert, Nicolaos Matsakis, and Pavel Veselý. Approximation Guarantees for Shortest Superstrings: Simpler and Better. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 29:1-29:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.29

Abstract

The Shortest Superstring problem is an NP-hard problem, in which given as input a set of strings, we are looking for a string of minimum length that contains all input strings as substrings. The Greedy Conjecture (Tarhio and Ukkonen, 1988) states that the GREEDY algorithm, which repeatedly merges the two strings of maximum overlap, is 2-approximate. We have recently shown (STOC 2022) that the approximation guarantee of GREEDY is at most (13+√{57})/6 ≈ 3.425. Before that, the best established upper bound for this was 3.5 by Kaplan and Shafrir (IPL 2005), which improved upon the upper bound of 4 by Blum et al. (STOC 1991). To derive our previous result, we established two incomparable upper bounds on the overlap sum of all cycle-closing edges in an optimal cycle cover and utilized lemmas of Blum et al. We improve the more involved one of the two bounds and, at the same time, make its proof more straightforward. This results in an improved approximation guarantee of (√{67}+2)/3 ≈ 3.396 for GREEDY. Additionally, our result implies an algorithm for the Shortest Superstring problem having an approximation guarantee of (√{67}+14)/9 ≈ 2.466, improving slightly upon the previously best guarantee of (√{57}+37)/18 ≈ 2.475 (STOC 2022).

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Shortest Superstring problem
  • Approximation Algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Chris Armen and Clifford Stein. Improved length bounds for the shortest superstring problem. In Proceedings of the 4th International Workshop on Algorithms and Data Structures (WADS), pages 494-505, 1995. URL: https://doi.org/10.1007/3-540-60220-8_88.
  2. Chris Armen and Clifford Stein. A 2 2/3 superstring approximation algorithm. Discret. Appl. Math., 88(1-3):29-57, 1998. URL: https://doi.org/10.1016/S0166-218X(98)00065-1.
  3. Avrim Blum, Tao Jiang, Ming Li, John Tromp, and Mihalis Yannakakis. Linear approximation of shortest superstrings. Journal of the ACM, 41(4):630-647, 1994. URL: https://doi.org/10.1145/179812.179818.
  4. Dany Breslauer, Tao Jiang, and Zhigen Jiang. Rotations of periodic strings and short superstrings. J. Algorithms, 24(2):340-353, 1997. URL: https://doi.org/10.1006/jagm.1997.0861.
  5. M. Crochemore and W. Rytter. Text Algorithms. Oxford University Press, 1994. Google Scholar
  6. Artur Czumaj, Leszek Gasieniec, Marek Piotrów, and Wojciech Rytter. Sequential and parallel approximation of shortest superstrings. J. Algorithms, 23(1):74-100, 1997. URL: https://doi.org/10.1006/jagm.1996.0823.
  7. Matthias Englert, Nicolaos Matsakis, and Pavel Veselý. Improved approximation guarantees for shortest superstrings using cycle classification by overlap to length ratios. In Proceedings of the 54th ACM Symposium on Theory of Computing (STOC), pages 317-330. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520001.
  8. Alan M. Frieze and Wojciech Szpankowski. Greedy algorithms for the shortest common superstring that are asymptotically optimal. Algorithmica, 21(1):21-36, 1998. Google Scholar
  9. Dan Gusfield. Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology. Cambridge University Press, 1997. URL: https://doi.org/10.1017/CBO9780511574931.
  10. Haim Kaplan, Moshe Lewenstein, Nira Shafrir, and Maxim Sviridenko. Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs. Journal of the ACM, 52(4):602-626, 2005. URL: https://doi.org/10.1145/1082036.1082041.
  11. Haim Kaplan and Nira Shafrir. The greedy algorithm for shortest superstrings. Inf. Process. Lett., 93(1):13-17, 2005. URL: https://doi.org/10.1016/j.ipl.2004.09.012.
  12. Marek Karpinski and Richard Schmied. Improved inapproximability results for the shortest superstring and related problems. In Proceedings of the 19th Computing: The Australasian Theory Symposium (CATS), pages 27-36, 2013. Google Scholar
  13. S. Rao Kosaraju, James K. Park, and Clifford Stein. Long tours and short superstrings. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science (FOCS), pages 166-177, 1994. URL: https://doi.org/10.1109/SFCS.1994.365696.
  14. Bin Ma. Why greed works for shortest common superstring problem. Theor. Comput. Sci., 410(51):5374-5381, 2009. URL: https://doi.org/10.1016/j.tcs.2009.09.014.
  15. Marcin Mucha. A tutorial on shortest superstring approximation. https://www.mimuw.edu.pl/~mucha/teaching/aa2008/ss.pdf, 2007. [Accessed 15-June-2023].
  16. Marcin Mucha. Lyndon words and short superstrings. In Proceedings of the 24th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 958-972, 2013. URL: https://doi.org/10.1137/1.9781611973105.69.
  17. Katarzyna Paluch, Khaled Elbassioni, and Anke van Zuylen. Simpler approximation of the maximum asymmetric traveling salesman problem. In Proceedings of the 29th Symposium on Theoretical Aspects of Computer Science (STACS), pages 501-506, 2012. URL: https://doi.org/10.4230/LIPIcs.STACS.2012.501.
  18. Steven Skiena. The Algorithm Design Manual, Third Edition. Texts in Computer Science. Springer, 2020. Google Scholar
  19. Ondřej Sladký, Pavel Veselý, and Karel Břinda. Masked superstrings as a unified framework for textual k-mer set representations. bioRxiv, 2023. URL: https://doi.org/10.1101/2023.02.01.526717.
  20. Z. Sweedyk. A 2onehalf-approximation algorithm for shortest superstring. SIAM J. Comput., 29(3):954-986, 1999. URL: https://doi.org/10.1137/S0097539796324661.
  21. Jorma Tarhio and Esko Ukkonen. A greedy approximation algorithm for constructing shortest common superstrings. Theor. Comput. Sci., 57:131-145, 1988. URL: https://doi.org/10.1016/0304-3975(88)90167-3.
  22. Shang-Hua Teng and Frances Yao. Approximating shortest superstrings. SIAM Journal on Computing, 26(2):410-417, 1997. URL: https://doi.org/10.1137/S0097539794286125.
  23. Jonathan S. Turner. Approximation algorithms for the shortest common superstring problem. Inf. Comput., 83(1):1-20, 1989. URL: https://doi.org/10.1016/0890-5401(89)90044-8.
  24. Virginia Vassilevska. Explicit inapproximability bounds for the shortest superstring problem. In 30th International Symposium, MFCS, Gdansk, Poland, volume 3618 of Lecture Notes in Computer Science, pages 793-800. Springer, 2005. Google Scholar
  25. Vijay Vazirani. Approximation algorithms. Springer, 2001. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail