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Recognizing Unit Multiple Intervals Is Hard

Authors Virginia Ardévol Martínez , Romeo Rizzi , Florian Sikora , Stéphane Vialette

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Graph theory
Keywords
  • Interval graphs
  • unit multiple interval graphs
  • recognition
  • NP-hardness

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Abstract

Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A d-interval is the union of d intervals on the real line, and a graph is a d-interval graph if it is the intersection graph of d-intervals. In particular, it is a unit d-interval graph if it admits a d-interval representation where every interval has unit length.
Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is NP-complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also NP-complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit d-interval graphs for any d ⩾ 2, which does not follow directly in graph recognition problems -as an example, it took almost 20 years to close the gap between d = 2 and d > 2 for the recognition of d-track interval graphs. Our result has several implications, including that recognizing (x, …, x) d-interval graphs and depth r unit 2-interval graphs is NP-complete for every x ⩾ 11 and every r ⩾ 4.

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Virginia Ardévol Martínez, Romeo Rizzi, Florian Sikora, and Stéphane Vialette. Recognizing Unit Multiple Intervals Is Hard. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.8

Author Details

Virginia Ardévol Martínez
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France
Romeo Rizzi
  • Department of Computer Science, University of Verona, Italy
Florian Sikora
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France
Stéphane Vialette
  • LIGM, CNRS, Univ Gustave Eiffel, F77454 Marne-la-Vallée, France

Funding

This work was partially supported by the ANR project ANR-21-CE48-0022 ("S-EX-AP-PE-AL").

Acknowledgements

Part of this work was conducted when RR was an invited professor at Université Paris-Dauphine.

References

  1. Amotz Bar-Noy, Reuven Bar-Yehuda, Ari Freund, Joseph Naor, and Baruch Schieber. A unified approach to approximating resource allocation and scheduling. J. ACM, 48(5):1069-1090, 2001. URL: https://doi.org/10.1145/502102.502107.
  2. Reuven Bar-Yehuda, Magnús M. Halldórsson, Joseph Naor, Hadas Shachnai, and Irina Shapira. Scheduling Split Intervals. SIAM J. Comput., 36(1):1-15, 2006. URL: https://doi.org/10.1137/S0097539703437843.
  3. Kenneth P. Bogart and Douglas B. West. A short proof that 'proper = unit'. Discret. Math., 201(1-3):21-23, 1999. URL: https://doi.org/10.1016/S0012-365X(98)00310-0.
  4. Kellogg S. Booth and George S. Lueker. Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms. J. Comput. Syst. Sci., 13(3):335-379, 1976. URL: https://doi.org/10.1016/S0022-0000(76)80045-1.
  5. Ayelet Butman, Danny Hermelin, Moshe Lewenstein, and Dror Rawitz. Optimization problems in multiple-interval graphs. ACM Trans. Algorithms, 6(2), 2010. URL: https://doi.org/10.1145/1721837.1721856.
  6. Joel E. Cohen. Food Webs and Niche Space, volume 11 of Monographs in Population Biology. Princeton University Press, 1978. Google Scholar
  7. Derek G. Corneil, Stephan Olariu, and Lorna Stewart. The LBFS structure and recognition of interval graphs. SIAM J. Discret. Math., 23(4):1905-1953, 2010. URL: https://doi.org/10.1137/S0895480100373455.
  8. Margaret B. Cozzens. Higher and Multi-Dimensional Analogues of Interval Graphs. PhD thesis, Rutgers University, 1982. Google Scholar
  9. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  10. Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the Parameterized Complexity of Multiple-Interval Graph Problems. Theor. Comput. Sci., 410(1):53-61, 2009. URL: https://doi.org/10.1016/j.tcs.2008.09.065.
  11. Michael R. Fellows, Jan Kratochvíl, Matthias Middendorf, and Frank Pfeiffer. The complexity of induced minors and related problems. Algorithmica, 13(3):266-282, 1995. URL: https://doi.org/10.1007/BF01190507.
  12. Peter C. Fishburn. Interval Orders and Interval Graphs: A Study of Partially Ordered Sets. Wiley, 1985. Google Scholar
  13. Mathew C. Francis, Daniel Gonçalves, and Pascal Ochem. The maximum clique problem in multiple interval graphs. Algorithmica, 71(4):812-836, 2015. URL: https://doi.org/10.1007/s00453-013-9828-6.
  14. Peter Frankl and Hiroshi Maehara. Open-interval graphs versus closed-interval graphs. Discret. Math., 63(1):97-100, 1987. URL: https://doi.org/10.1016/0012-365X(87)90156-7.
  15. Philippe Gambette and Stéphane Vialette. On restrictions of balanced 2-interval graphs. In Andreas Brandstädt, Dieter Kratsch, and Haiko Müller, editors, Graph-Theoretic Concepts in Computer Science, 33rd International Workshop, WG 2007, Dornburg, Germany, June 21-23, 2007. Revised Papers, volume 4769 of LNCS, pages 55-65. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-74839-7_6.
  16. Jerrold R. Griggs and Douglas B. West. Extremal values of the interval number of a graph. SIAM J. Algebraic Discret. Methods, 1(1):1-7, 1980. URL: https://doi.org/10.1137/0601001.
  17. András Gyárfás and Douglas West. Multitrack interval graphs. Congressus Numerantium, pages 109-116, 1995. Google Scholar
  18. Minghui Jiang. Recognizing d-interval graphs and d-track interval graphs. Algorithmica, 66(3):541-563, 2013. URL: https://doi.org/10.1007/s00453-012-9651-5.
  19. Deborah Joseph, Joao Meidanis, and Prasoon Tiwari. Determining DNA sequence similarity using maximum independent set algorithms for interval graphs. In Otto Nurmi and Esko Ukkonen, editors, Algorithm Theory - SWAT '92, Third Scandinavian Workshop on Algorithm Theory, Helsinki, Finland, July 8-10, 1992, Proceedings, volume 621 of LNCS, pages 326-337. Springer, 1992. URL: https://doi.org/10.1007/3-540-55706-7_29.
  20. Jan Kratochvíl. A special planar satisfiability problem and a consequence of its NP-completeness. Discret. Appl. Math., 52(3):233-252, 1994. URL: https://doi.org/10.1016/0166-218X(94)90143-0.
  21. Cornelis Gerrit Lekkerkerker and Johan Ch. Boland. Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae, 51:45-64, 1962. Google Scholar
  22. Robert McGuigan. Presentation at NSF-CBMS Conference at Colby College, 1977. Google Scholar
  23. Terry A McKee and Fred R McMorris. Topics in intersection graph theory. SIAM, 1999. Google Scholar
  24. Dieter Rautenbach and Jayme L Szwarcfiter. Unit interval graphs of open and closed intervals. J. Graph Theory, 72(4):418-429, 2013. URL: https://doi.org/10.1002/jgt.21650.
  25. Fred S. Roberts. Indifference graphs. In F. Harary, editor, Proof Techniques in Graph Theory, pages 139-146. Academic Press, NY, 1969. Google Scholar
  26. Fred S. Roberts. On the boxicity and cubicity of a graph. In W. T. Tutte, editor, Recent Progress in Combinatorics, pages 301-310. Academic Press, NY, 1969. Google Scholar
  27. Fred S. Roberts. Graph theory and its applications to problems of society. SIAM, 1978. Google Scholar
  28. Alexandre Simon. Algorithmic study of 2-interval graphs. Master’s thesis, Delft University of Technology, 2021. Google Scholar
  29. William T. Trotter and Frank Harary. On double and multiple interval graphs. J. Graph Theory, 3(3):205-211, 1979. URL: https://doi.org/10.1002/jgt.3190030302.
  30. Stéphane Vialette. On the computational complexity of 2-interval pattern matching problems. Theor. Comput. Sci., 312(2-3):223-249, 2004. URL: https://doi.org/10.1016/j.tcs.2003.08.010.
  31. Douglas B. West and David B. Shmoys. Recognizing graphs with fixed interval number is NP-complete. Disrecte Appl. Math., 8:295-305, 1984. URL: https://doi.org/10.1016/0166-218X(84)90127-6.
  32. Mihalis Yannakakis. The complexity of the partial order dimension problem. SIAM Journal on Algebraic Discrete Methods, 3(3):351-358, 1982. Google Scholar
  33. Peisen Zhang, Eric A Schon, Stuart G Fischer, Eftihia Cayanis, Janie Weiss, Susan Kistler, and Philip E Bourne. An algorithm based on graph theory for the assembly of contigs in physical mapping of DNA. Comput. Appl. Biosci., 10(3):309-317, 1994. URL: https://doi.org/10.1093/bioinformatics/10.3.309.
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