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Interval Deletion Is Fixed-Parameter Tractable

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Published:13 January 2015Publication History
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Abstract

We study the minimum interval deletion problem, which asks for the removal of a set of at most k vertices to make a graph of n vertices into an interval graph. We present a parameterized algorithm of runtime 10knO(1) for this problem—that is, we show that the problem is fixed-parameter tractable.

References

  1. Isolde Adler, Martin Grohe, and Stephan Kreutzer. 2008. Computing excluded minors. In Proceedings of SODA 2008. 641--650. DOI:http://dx.doi.org/10.1145/1347082.1347153 Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Amotz Bar-Noy, Reuven Bar-Yehuda, Ari Freund, Joseph Naor, and Baruch Schieber. 2001. A unified approach to approximating resource allocation and scheduling. Journal of the ACM 48, 5, 1069--1090. DOI:http://dx.doi.org/10.1145/502102.502107 Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Seymour Benzer. 1959. On the topology of the genetic fine structure. Proceedings of the National Academy of Sciences 45, 11, 1607--1620.Google ScholarGoogle ScholarCross RefCross Ref
  4. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, Michael T. Hallett, and Harold T. Wareham. 1995. Parameterized complexity analysis in computational biology. Computer Applications in the Biosciences 11, 1, 49--57. DOI:http://dx.doi.org/10.1093/bioinformatics/11.1.49Google ScholarGoogle Scholar
  5. Kellogg S. Booth and George S. Lueker. 1976. Testing for the consecutive ones property, interval graphs, and graph planarity using P Q-tree algorithms. Journal of Computer and System Sciences 13, 3, 335--379. DOI:http://dx.doi.org/10.1016/S0022-0000(76)80045-1 A preliminary version appeared in STOC 1975. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Andreas Brandstädt, Van Bang Le, and Jeremy P. Spinrad. 1999. Graph Classes: A Survey. SIAM, Philadelphia, PA. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Peter Buneman. 1974. A characterization of rigid circuit graphs. Discrete Mathematics 9, 4, 205--212. DOI:http://dx.doi.org/10.1016/j.dam.2008.07.004 Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Leizhen Cai. 1996. Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters 58, 4, 171--176. DOI:http://dx.doi.org/10.1016/0020-0190(96)00050-6 Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Yixin Cao, Jianer Chen, and Yang Liu. 2010. On feedback vertex set: New measure and new structures. In Algorithm Theory—SWAT 2010. Lecture Notes in Computer Science, Vol. 6139. Springer, 93--104. DOI:http://dx.doi.org/10.1007/978-3-642-13731-0_10 Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Jianer Chen, Yang Liu, Songjian Lu, Barry O'Sullivan, and Igor Razgon. 2008. A fixed-parameter algorithm for the directed feedback vertex set problem. Journal of the ACM 55, 5, Article No. 21. DOI:http://dx.doi.org/10.1145/1411509.1411511 Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Gabriel A. Dirac. 1961. On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 25, 1, 71--76. DOI:http://dx.doi.org/10.1007/BF02992776Google ScholarGoogle Scholar
  12. Rodney G. Downey and Michael R. Fellows. 2013. Fundamentals of Parameterized Complexity. Springer. DOI:http://dx.doi.org/10.1007/978-1-4471-5559-1 Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. 2012. Planar F-deletion: Approximation and optimal FPT algorithms. In Proceedings of FOCS 2012. IEEE, Los Alamitos, CA, 470--479. DOI:http://dx.doi.org/10.1109/FOCS.2012.62 Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Delbert R. Fulkerson and Oliver A. Gross. 1965. Incidence matrices and interval graphs. Pacific Journal of Mathematics 15, 3, 835--855.Google ScholarGoogle ScholarCross RefCross Ref
  15. Tibor Gallai. 1967. Transitiv orientierbare Graphen. Acta Mathematica Academiae Scientiarum Hungarica 18, 1--2, 25--66. DOI:http://dx.doi.org/10.1007/BF02020961. Translated by Frédéric Maffray and Myriam Preissmann in Perfect Graphs, Jorge L. Ramírez-Alfonsín and Bruce A. Reed (Eds.). Wiley, 2001, 25--66.Google ScholarGoogle ScholarCross RefCross Ref
  16. Paul W. Goldberg, Martin C. Golumbic, Haim Kaplan, and Ron Shamir. 1995. Four strikes against physical mapping of DNA. Journal of Computational Biology 2, 1, 139--152. DOI:http://dx.doi.org/10.1089/ cmb.1995.2.139Google ScholarGoogle ScholarCross RefCross Ref
  17. Martin C. Golumbic. 2004. Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, Vol. 57. North-Holland Publishing, Amsterdam, The Netherlands. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. György Hajós. 1957. (Problem 65) Über eine Art von Graphen. Internationale Mathematische Nachrichten 11.Google ScholarGoogle Scholar
  19. Haim Kaplan, Ron Shamir, and Robert E. Tarjan. 1999. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM Journal on Computing 28, 5, 1906--1922. DOI:http://dx.doi.org/10.1137/S0097539796303044 A preliminary version appeared in FOCS 1994. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Richard M. Karp. 1993. Mapping the genome: Some combinatorial problems arising in molecular biology. In Proceedings of STOC 1993 ACM, New York, NY, 278--285. DOI:http://dx.doi.org/10.1145/167088.167170 Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Ken-ichi Kawarabayashi. 2009. Planarity allowing few error vertices in linear time. In Proceedings of FOCS 2009. IEEE, Los Alamitos, CA, 639--648. DOI:http://dx.doi.org/10.1109/FOCS.2009.45 Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Ken-ichi Kawarabayashi and Bruce A. Reed. 2010. An (almost) linear time algorithm for odd cyles transversal. In Proceedings of SODA 2010.365--378. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. David George Kendall. 1969. Incidence matrices, interval graphs and seriation in archaeology. Pacific Journal of Mathematics 28, 565--570.Google ScholarGoogle ScholarCross RefCross Ref
  24. Ton Kloks, Dieter Kratsch, and C. K. Wong. 1998. Minimum fill-in on circle and circular-arc graphs. Journal of Algorithms 28, 2, 272--289. DOI:http://dx.doi.org/10.1006/jagm.1998.0936 A preliminary version appeared in ICALP 1996. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. C. G. Lekkerkerker and J. Ch. Boland. 1962. Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae 51, 45--64.Google ScholarGoogle ScholarCross RefCross Ref
  26. John M. Lewis and Mihalis Yannakakis. 1980. The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences 20, 2, 219--230. DOI:http://dx.doi.org/10.1016/0022-0000(80)90060-4. Preliminary versions were independently presented in STOC 1978.Google ScholarGoogle ScholarCross RefCross Ref
  27. Carsten Lund and Mihalis Yannakakis. 1993. The approximation of maximum subgraph problems. In Automata, Languages, and Programming. Lecture Notes in Computer Science, Vol. 700. Springer, 40--51. DOI:http://dx.doi.org/10.1007/3-540-56939-1_60 Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. Dániel Marx. 2010. Chordal deletion is fixed-parameter tractable. Algorithmica 57, 4, 747--768. DOI:http://dx.doi.org/10.1007/s00453-008-9233-8 Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. Dániel Marx and Ildikó Schlotter. 2012. Obtaining a planar graph by vertex deletion. Algorithmica 62, 3--4, 807--822. DOI:http://dx.doi.org/10.1007/s00453-010-9484-z Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. N. S. Narayanaswamy and R. Subashini. 2013. FPT algorithms for consecutive ones submatrix problems. In Parameterized and Exact Computation. Lecture Notes in Computer Science, Vol. 8246. Springer, 295--307. DOI:http://dx.doi.org/10.1007/978-3-319-03898-8_25Google ScholarGoogle Scholar
  31. Sheng-Lung Peng and Chi-Kang Chen. 2006. On the interval completion of chordal graphs. Discrete Applied Mathematics 154, 6, 1003--1010. DOI:http://dx.doi.org/10.1016/j.dam.2005.09.010 Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. Bruce A. Reed, Kaleigh Smith, and Adrian Vetta. 2004. Finding odd cycle transversals. Operations Research Letters 32, 4, 299--301. DOI:http://dx.doi.org/10.1016/j.orl.2003.10.009 Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. Pim Van't Hof and Yngve Villanger. 2013. Proper interval vertex deletion. Algorithmica 65, 4, 845--867. DOI:http://dx.doi.org/10.1007/s00453-012-9661-3Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. Yngve Villanger, Pinar Heggernes, Christophe Paul, and Jan Arne Telle. 2009. Interval completion is fixed parameter tractable. SIAM Journal on Computing 38, 5, 2007--2020. DOI:http://dx.doi.org/10.1137/070710913. A preliminary version appeared in STOC 2007. Google ScholarGoogle ScholarDigital LibraryDigital Library

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        cover image ACM Transactions on Algorithms
        ACM Transactions on Algorithms  Volume 11, Issue 3
        January 2015
        269 pages
        ISSN:1549-6325
        EISSN:1549-6333
        DOI:10.1145/2721890
        Issue’s Table of Contents

        Copyright © 2015 ACM

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        Publication History

        • Published: 13 January 2015
        • Accepted: 1 April 2014
        • Revised: 1 January 2014
        • Received: 1 May 2013
        Published in talg Volume 11, Issue 3

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