Document Open Access Logo

The Simultaneous Interval Number: A New Width Parameter that Measures the Similarity to Interval Graphs

Authors Jesse Beisegel , Nina Chiarelli , Ekkehard Köhler, Martin Milanič , Peter Muršič , Robert Scheffler

PDF

File

Thumbnail PDF
PDF
LIPIcs.SWAT.2024.7.pdf
  • Filesize: 0.81 MB
  • 20 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Interval graph
  • simultaneous representation
  • width parameter
  • algorithm
  • parameterized complexity

Metrics

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

Abstract

We propose a novel way of generalizing the class of interval graphs, via a graph width parameter called simultaneous interval number. This parameter is related to the simultaneous representation problem for interval graphs and defined as the smallest number d of labels such that the graph admits a d-simultaneous interval representation, that is, an assignment of intervals and label sets to the vertices such that two vertices are adjacent if and only if the corresponding intervals, as well as their label sets, intersect. We show that this parameter is NP-hard to compute and give several bounds for the parameter, showing in particular that it is sandwiched between pathwidth and linear mim-width. For classes of graphs with bounded parameter values, assuming that the graph is equipped with a simultaneous interval representation with a constant number of labels, we give FPT algorithms for the clique, independent set, and dominating set problems, and hardness results for the independent dominating set and coloring problems. The FPT results for independent set and dominating set are for the simultaneous interval number plus solution size. In contrast, both problems are known to be 𝖶[1]-hard for linear mim-width plus solution size.

Cite As Get BibTex

Jesse Beisegel, Nina Chiarelli, Ekkehard Köhler, Martin Milanič, Peter Muršič, and Robert Scheffler. The Simultaneous Interval Number: A New Width Parameter that Measures the Similarity to Interval Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SWAT.2024.7

Author Details

Jesse Beisegel
  • Institute of Mathematics, Brandenburg University of Technology, Cottbus, Germany
Nina Chiarelli
  • FAMNIT and IAM, University of Primorska, Koper, Slovenia
Ekkehard Köhler
  • Institute of Mathematics, Brandenburg University of Technology, Cottbus, Germany
Martin Milanič
  • FAMNIT and IAM, University of Primorska, Koper, Slovenia
Peter Muršič
  • FAMNIT, University of Primorska, Koper, Slovenia
Robert Scheffler
  • Institute of Mathematics, Brandenburg University of Technology, Cottbus, Germany

Funding

The authors acknowledge partial support by the Slovenian Research and Innovation Agency (I0-0035, research programs P1-0285 and P1-0404, and research projects N1-0160, N1-0210, J1-3001, J1-3002, J1-3003, J1-4008, and J1-4084) and by the research program CogniCom (0013103) at the University of Primorska.

Acknowledgements

The authors would like to thank the reviewers for their helpful remarks; in particular, for their hints on thinness and boxicity.

References

  1. Jesse Beisegel, Nina Chiarelli, Ekkehard Köhler, Martin Milanič, Peter Muršič, and Robert Scheffler. The simultaneous interval number: A new width parameter that measures the similarity to interval graphs, 2024. URL: https://arxiv.org/abs/2404.10670.
  2. Rémy Belmonte and Martin Vatshelle. Graph classes with structured neighborhoods and algorithmic applications. Theoretical Computer Science, 511:54-65, 2013. URL: https://doi.org/10.1016/j.tcs.2013.01.011.
  3. Alan A. Bertossi. Dominating sets for split and bipartite graphs. Information Processing Letters, 19(1):37-40, 1984. URL: https://doi.org/10.1016/0020-0190(84)90126-1.
  4. Thomas Bläsius and Ignaz Rutter. Simultaneous PQ-ordering with applications to constrained embedding problems. ACM Transactions on Algorithms, 12(2):46, 2016. Id/No 16. URL: https://doi.org/10.1145/2738054.
  5. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing, 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  6. Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209(1-2):1-45, 1998. URL: https://doi.org/10.1016/S0304-3975(97)00228-4.
  7. Jan Bok and Nikola Jedličková. A note on simultaneous representation problem for interval and circular-arc graphs, 2018. URL: https://arxiv.org/abs/1811.04062.
  8. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width. I: Tractable FO model checking. Journal of the ACM, 69(1):3:1-3:46, 2022. URL: https://doi.org/10.1145/3486655.
  9. Flavia Bonomo and Diego de Estrada. On the thinness and proper thinness of a graph. Discrete Applied Mathematics, 261:78-92, 2019. URL: https://doi.org/10.1016/j.dam.2018.03.072.
  10. Flavia Bonomo-Braberman and Gastón Abel Brito. Intersection models and forbidden pattern characterizations for 2-thin and proper 2-thin graphs. Discrete Applied Mathematics, 339:53-77, 2023. URL: https://doi.org/10.1016/j.dam.2023.06.013.
  11. Flavia Bonomo-Braberman, Carolina L. Gonzalez, Fabiano S. Oliveira, Moysés S. Sampaio Jr., and Jayme L. Szwarcfiter. Thinness of product graphs. Discrete Applied Mathematics, 312:52-71, 2022. URL: https://doi.org/10.1016/j.dam.2021.04.003.
  12. Kellogg S. Booth and J. Howard Johnson. Dominating sets in chordal graphs. SIAM Journal on Computing, 11:191-199, 1982. URL: https://doi.org/10.1137/0211015.
  13. Sunil Chandran, Carlo Mannino, and Gianpaolo Oriolo. The indepedent set problem and the thinness of a graph. Unpublished manuscript cited in [11], 2007. Google Scholar
  14. Steven Chaplick, Martin Töpfer, Jan Voborník, and Peter Zeman. On H-topological intersection graphs. Algorithmica, 83(11):3281-3318, 2021. URL: https://doi.org/10.1007/s00453-021-00846-3.
  15. Carlo Comin and Romeo Rizzi. An improved upper bound on maximal clique listing via rectangular fast matrix multiplication. Algorithmica, 80(12):3525-3562, 2018. URL: https://doi.org/10.1007/s00453-017-0402-5.
  16. Derek G. Corneil and Yehoshua Perl. Clustering and domination in perfect graphs. Discrete Applied Mathematics, 9(1):27-39, 1984. URL: https://doi.org/10.1016/0166-218X(84)90088-X.
  17. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Cham: Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  18. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. I: Graph classes with a forbidden structure. SIAM Journal on Discrete Mathematics, 35(4):2618-2646, 2021. URL: https://doi.org/10.1137/20M1352119.
  19. Clément Dallard, Martin Milanič, and Kenny Štorgel. Treewidth versus clique number. II: Tree-independence number. Journal of Combinatorial Theory. Series B, 164:404-442, 2024. URL: https://doi.org/10.1016/j.jctb.2023.10.006.
  20. Clément Dallard, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, and Martin Milanič. Computing tree decompositions with small independence number. In Karl Bringmann, Martin Grohe, Gabriele Puppis, and Ola Svensson, editors, 51th International Colloquium on Automata, Languages, and Programming, ICALP 2024, July 8-12, 2024, Tallinn, Estonia, volume 297 of LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. To appear. Google Scholar
  21. H. N. de Ridder et al. Entry "2-interval" in Information System on Graph Classes and their Inclusions (ISGCI). URL: https://graphclasses.org/classes/gc_40.html.
  22. H. N. de Ridder et al. Entry "3-track" in Information System on Graph Classes and their Inclusions (ISGCI). URL: https://graphclasses.org/classes/gc_1080.html.
  23. H. N. de Ridder et al. Entry "3K₁-free" in Information System on Graph Classes and their Inclusions (ISGCI). URL: https://graphclasses.org/classes/AUTO_399.html.
  24. Michael R. Fellows, Danny Hermelin, Frances Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science, 410(1):53-61, 2009. URL: https://doi.org/10.1016/j.tcs.2008.09.065.
  25. Fedor V. Fomin, Petr A. Golovach, and Jean-Florent Raymond. On the tractability of optimization problems on H-graphs. Algorithmica, 82(9):2432-2473, 2020. URL: https://doi.org/10.1007/s00453-020-00692-9.
  26. Mathew C. Francis, Daniel Gonçalves, and Pascal Ochem. The maximum clique problem in multiple interval graphs. Algorithmica, 71:812-836, 2015. URL: https://doi.org/10.1007/s00453-013-9828-6.
  27. Delbert R. Fulkerson and Oliver A. Gross. Incidence matrices and interval graphs. Pacific Journal of Mathematics, 15:835-855, 1965. URL: https://doi.org/10.2140/pjm.1965.15.835.
  28. Michael R. Garey, David S. Johnson, Gerald L. Miller, and Christos H. Papadimitriou. The complexity of coloring circular arcs and chords. SIAM Journal on Algebraic Discrete Methods, 1(2):216-227, 1980. URL: https://doi.org/10.1137/0601025.
  29. Fănică Gavril. Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM Journal on Computing, 1(2):180-187, 1972. URL: https://doi.org/10.1137/0201013.
  30. Petr A. Golovach, Pinar Heggernes, Mamadou Moustapha Kanté, Dieter Kratsch, Sigve H. Sæther, and Yngve Villanger. Output-polynomial enumeration on graphs of bounded (local) linear MIM-width. Algorithmica, 80(2):714-741, 2018. URL: https://doi.org/10.1007/s00453-017-0289-1.
  31. Martin Charles Golumbic and Udi Rotics. On the clique-width of some perfect graph classes. International Journal of Foundations of Computer Science, 11(3):423-443, 2000. URL: https://doi.org/10.1142/S0129054100000260.
  32. Carolina Lucía Gonzalez and Felix Mann. On d-stable locally checkable problems parameterized by mim-width. Discrete Applied Mathematics, 347:1-22, 2024. URL: https://doi.org/10.1016/j.dam.2023.12.015.
  33. David A. Gregory and Norman J. Pullman. On a clique covering problem of Orlin. Discrete Mathematics, 41(1):97-99, 1982. URL: https://doi.org/10.1016/0012-365X(82)90085-1.
  34. Jerrold R. Griggs and Douglas B. West. Extremal values of the interval number of a graph. SIAM Journal on Algebraic Discrete Methods, 1(1):1-7, 1980. URL: https://doi.org/10.1137/0601001.
  35. Udaiprakash I. Gupta, Der-Tsai Lee, and Joseph Y.-T. Leung. An optimal solution for the channel-assignment problem. IEEE Transactions on Computers, C-28(11):807-810, 1979. URL: https://doi.org/10.1109/TC.1979.1675260.
  36. András Gyárfás and Douglas B. West. Multitrack interval graphs. In Proceedings of the Twenty-sixth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1995), volume 109 of Congressus Numerantium, pages 109-116, 1995. Google Scholar
  37. Akihiro Hashimoto and James Stevens. Wire routing by optimizing channel assignment within large apertures. In Proceedings of the 8th Design Automation Workshop, DAC '71, pages 155-169, New York, NY, USA, 1971. Association for Computing Machinery. URL: https://doi.org/10.1145/800158.805069.
  38. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. Special issue on the Fourteenth Annual IEEE Conference on Computational Complexity (Atlanta, GA, 1999). URL: https://doi.org/10.1006/jcss.2000.1727.
  39. Lars Jaffke, O-joung Kwon, Torstein J. F. Strømme, and Jan Arne Telle. Mim-width. III. Graph powers and generalized distance domination problems. Theoretical Computer Science, 796:216-236, 2019. URL: https://doi.org/10.1016/j.tcs.2019.09.012.
  40. Lars Jaffke, O-joung Kwon, and Jan Arne Telle. Mim-width. I. Induced path problems. Discrete Applied Mathematics, 278:153-168, 2020. URL: https://doi.org/10.1016/j.dam.2019.06.026.
  41. Krishnam Raju Jampani and Anna Lubiw. The simultaneous representation problem for chordal, comparability and permutation graphs. In Frank K. H. A. Dehne andMarina L. Gavrilova, Jörg-Rüdiger Sack, and Csaba D. Tóth, editors, Algorithms and Data Structures, 11th International Symposium, WADS 2009, Banff, Canada, August 21-23, 2009. Proceedings, volume 5664 of Lecture Notes in Computer Science, pages 387-398. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-03367-4_34.
  42. Krishnam Raju Jampani and Anna Lubiw. Simultaneous interval graphs. In Otfried Cheong, Kyung-Yong Chwa, and Kunsoo Park, editors, Algorithms and Computation - 21st International Symposium, ISAAC 2010, Jeju Island, Korea, December 15-17, 2010, Proceedings, Part I, volume 6506 of Lecture Notes in Computer Science, pages 206-217. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-17517-6_20.
  43. Krishnam Raju Jampani and Anna Lubiw. The simultaneous representation problem for chordal, comparability and permutation graphs. Journal of Graph Algorithms and Applications, 16(2):283-315, 2012. URL: https://doi.org/10.7155/jgaa.00259.
  44. Minghui Jiang. On the parameterized complexity of some optimization problems related to multiple-interval graphs. Theoretical Computer Science, 411(49):4253-4262, 2010. URL: https://doi.org/10.1016/j.tcs.2010.09.001.
  45. 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.
  46. J. Mark Keil. Finding Hamiltonian circuits in interval graphs. Information Processing Letters, 20:201-206, 1985. URL: https://doi.org/10.1016/0020-0190(85)90050-X.
  47. Brian W. Kernighan, Daniel G. Schweikert, and G. Persky. An optimum channel-routing algorithm for polycell layouts of integrated circuits. In J. Michael Galey, Herbert M. Wall, Robert B. Hitchcock Sr., Ben E. Britt, Richard E. Merwin, Donald J. Humcke, and David B. Smithhisler, editors, Proceedings of the 10th Design Automation Workshop, DAC '73, Portland, Oregon, USA, June 25-27, 1973, pages 50-59. ACM, 1973. URL: https://doi.org/10.1145/62882.62886.
  48. Tuukka Korhonen. A single-exponential time 2-approximation algorithm for treewidth. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science - FOCS 2021, pages 184-192. IEEE Computer Soc., Los Alamitos, CA, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00026.
  49. Chunmei Liu and Yinglei Song. Parameterized complexity and inapproximability of dominating set problem in chordal and near chordal graphs. Journal of combinatorial optimization, 22(4):684-698, 2011. URL: https://doi.org/10.1007/s10878-010-9317-7.
  50. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bulletin of EATCS, 105:41-71, 2011. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/92.
  51. Carlo Mannino, Gianpaolo Oriolo, Federico Ricci, and Sunil Chandran. The stable set problem and the thinness of a graph. Operations Research Letters, 35(1):1-9, 2007. URL: https://doi.org/10.1016/j.orl.2006.01.009.
  52. Dániel Marx. A short proof of the NP-completeness of circular arc coloring. Unpublished manuscript, 2003. URL: https://www.cs.bme.hu/~dmarx/papers/circularNP.pdf.
  53. Martin Milanič and Paweł Rzążewski. Tree decompositions with bounded independence number: beyond independent sets, 2022. URL: https://arxiv.org/abs/2209.12315.
  54. Sang-il Oum and Paul Seymour. Approximating clique-width and branch-width. Journal of Combinatorial Theory. Series B, 96(4):514-528, 2006. URL: https://doi.org/10.1016/j.jctb.2005.10.006.
  55. Boram Park, Suh-Ryung Kim, and Yoshio Sano. The competition numbers of complete multipartite graphs and mutually orthogonal Latin squares. Discrete Mathematics, 309(23-24):6464-6469, 2009. URL: https://doi.org/10.1016/j.disc.2009.06.016.
  56. Krzysztof Pietrzak. On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. Journal of Computer and System Sciences, 67(4):757-771, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00078-3.
  57. Ganesan Ramalingam and C. Pandu Rangan. A unified approach to domination problems on interval graphs. Information Processing Letters, 27(5):271-274, 1988. URL: https://doi.org/10.1016/0020-0190(88)90091-9.
  58. Ignaz Rutter, Darren Strash, Peter Stumpf, and Michael Vollmer. Simultaneous representation of proper and unit interval graphs. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, volume 144 of LIPIcs, pages 80:1-80:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.ESA.2019.80.
  59. Shuji Tsukiyama, Mikio Ide, Hiromu Ariyoshi, and Isao Shirakawa. A new algorithm for generating all the maximal independent sets. SIAM Journal on Computing, 6(3):505-517, 1977. URL: https://doi.org/10.1137/0206036.
  60. Martin Vatshelle. New Width Parameters of Graphs. PhD thesis, University of Bergen, 2012. URL: https://hdl.handle.net/1956/6166.
  61. Jens Vygen. NP-completeness of some edge-disjoint paths problems. Discrete Applied Mathematics, 61(1):83-90, 1995. URL: https://doi.org/10.1016/0166-218X(93)E0177-Z.
  62. Nikola Yolov. Minor-matching hypertree width. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 219-233. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.16.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact