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Extending Partial Representations of Proper and Unit Interval Graphs

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Abstract

The recently introduced problem of extending partial interval representations asks, for an interval graph with some intervals pre-drawn by the input, whether the partial representation can be extended to a representation of the entire graph. In this paper, we give a linear-time algorithm for extending proper interval representations and an almost quadratic-time algorithm for extending unit interval representations. We also introduce the more general problem of bounded representations of unit interval graphs, where the input constrains the positions of some intervals by lower and upper bounds. We show that this problem is NP-complete for disconnected input graphs and give a polynomial-time algorithm for the special class of instances, where the ordering of the connected components of the input graph along the real line is prescribed. This includes the case of partial representation extension. The hardness result sharply contrasts the recent polynomial-time algorithm for bounded representations of proper interval graphs (Balko et al. in 2013). So unless \({\textsf {P}} = {\textsf {NP}}\), proper and unit interval representations have vastly different structure. This explains why partial representation extension problems for these different types of representations require substantially different techniques.

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Notes

  1. The complexity depends greatly on the considered computational model. If the arithmetic machine model is used (where the cost of all arithmetic operations is constant, independent of the size of operands), then this algorithm runs in time \(\mathcal {O}{}(n^2)\).

  2. Notice that, in the partial representation, some intervals may share position. But if two endpoints \(\ell _i\) and \(r_j\) share the position, then \(v_iv_j \in E(G)\) and we break the tie by setting \(\ell _i \lessdot r_j\).

  3. If \(\varepsilon \) was not of the form \(1 \over K\), then the grid could not contain both left and right endpoints of the intervals. We reserve K for the value \(1 \over \varepsilon \) in this paper.

  4. In other words, for the smallest shifts we assign the right-shift 0; for the second smallest shifts, we assign \(\varepsilon \); for the third smallest shifts, \(2\varepsilon \); and so on.

References

  1. Angelini, P., Battista, G.D., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. ACM Trans. Algorithms 11(4), 32:1–32:42 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balko, M., Klavík, P., Otachi, Y.: Bounded representations of interval and proper interval graphs. In: Algorithms and Computation—24th International Symposium, ISAAC 2013, Lecture Notes in Computer Science, vol. 8283, pp. 535–546. Springer, Berlin (2013)

  3. Balof, B., Doignon, J.P., Fiorini, S.: The representation polyhedron of a semiorder. Order 30(1), 103–135 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bang-Jensen, J., Huang, J., Zhu, X.: Completing Orientations of Partially Oriented Graphs. CoRR (2015). arXiv:1509.01301

  5. Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. In: SODA’13: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1030–1043 (2013)

  6. Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13, 335–379 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chaplick, S., Dorbec, P., Kratochvíl, J., Montassier, M., Stacho, J.: Contact representations of planar graphs: extending a partial representation is hard. In: Graph-Theoretic Concepts in Computer Science—40th International Workshop, WG 2014, Lecture Notes in Computer Science, vol. 8747, pp. 139–151. Springer, Berlin (2014)

  8. Chaplick, S., Fulek, R., Klavík, P.: Extending partial representations of circle graphs. In: Graph Drawing—21st International Symposium, GD 2013, Lecture Notes in Computer Science, vol. 8242, pp. 131–142. Springer, Berlin (2013)

  9. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)

    MATH  Google Scholar 

  10. Corneil, D.G., Kim, H., Natarajan, S., Olariu, S., Sprague, A.P.: Simple linear time recognition of unit interval graphs. Inf. Process. Lett. 55(2), 99–104 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Corneil, D.G., Olariu, S., Stewart, L.: The LBFS structure and recognition of interval graphs. SIAM J. Discret. Math. 23(4), 1905–1953 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Deng, X., Hell, P., Huang, J.: Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs. SIAM J. Comput. 25(2), 390–403 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Durán, G., Fernández Slezak, F., Grippo, L.N., Oliveira, F.d.S., Szwarcfiter, J.: On unit interval graphs with integer endpoints. In: LAGOS 2015 (2015). http://www.lia.ufc.br/lagos2015/docs/endm/ENDM50_74.pdf

  14. Fürer, M.: Faster integer multiplication. SIAM J. Comput. 39(3), 979–1005 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Garey, M.R., Johnson, D.S.: Complexity results for multiprocessor scheduling under resource constraints. SIAM J. Comput. 4(4), 397–411 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and of interval graphs. Can. J. Math. 16, 539–548 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  17. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. North-Holland, Amsterdam (2004)

    MATH  Google Scholar 

  18. Hajós, G.: Über eine Art von Graphen. Int. Math. Nachr. 11, 65 (1957)

    Google Scholar 

  19. Jampani, K.R., Lubiw, A.: Simultaneous interval graphs. In: Algorithms and Computation—21st International Symposium, ISAAC 2010, Lecture Notes in Computer Science, vol. 6506, pp. 206–217. Springer, Berlin (2010)

  20. Jampani, K.R., Lubiw, A.: The simultaneous representation problem for chordal, comparability and permutation graphs. J. Graph Algorithms Appl. 16(2), 283–315 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Klavík, P., Kratochvíl, J., Krawczyk, T., Walczak, B.: Extending partial representations of function graphs and permutation graphs. In: Algorithms—20th Annual European Symposium, ESA 2012, Lecture Notes in Computer Science, vol. 7501, pp. 671–682. Springer, Berlin (2012)

  23. Klavík, P., Kratochvíl, J., Otachi, Y., Rutter, I., Saitoh, T., Saumell, M., Vyskočil, T.: Extending partial representations of proper and unit interval graphs. In: Algorithm Theory—14th Scandinavian Symposium and Workshops, SWAT 2014, Lecture Notes in Computer Science, vol. 8503, pp. 253–264. Springer, Berlin (2014)

  24. Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T.: Extending partial representations of subclasses of chordal graphs. Theor. Comput. Sci. 576, 85–101 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T., Vyskocil, T.: Extending partial representations of interval graphs. CoRR (2013). arXiv:1306.2182

  26. Klavík, P., Kratochvíl, J., Vyskočil, T.: Extending partial representations of interval graphs. In: Theory and Applications of Models of Computation—8th Annual Conference, TAMC 2011, Lecture Notes in Computer Science, vol. 6648, pp. 276–285. Springer, Berlin (2011)

  27. Klavík, P., Otachi, Y., Šejnoha, J.: On the classes of interval graphs of limited nesting and count of lengths. CoRR (2015). arXiv:1510.03998

  28. Klavík, P., Saumell, M.: Minimal obstructions for partial representations of interval graphs. In: Algorithms and Computation—25th International Symposium, ISAAC 2014, Lecture Notes in Computer Science, vol. 8889, pp. 401–413. Springer, Berlin (2014)

  29. Patrignani, M.: On extending a partial straight-line drawing. Int. J. Found. Comput. Sci. 17(5), 1061–1070 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pirlot, M.: Minimal representation of a semiorder. Theory Decis. 28, 109–141 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pirlot, M.: Synthetic description of a semiorder. Discret. Appl. Math. 31(3), 299–308 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. Roberts, F.S.: Representations of Indifference Relations. Ph.D. Thesis, Stanford University (1968)

  33. Roberts, F.S.: Indifference graphs. In: Harary, F. (ed.) Proof Techniques in Graph Theory, pp. 139–146. Academic Press, London (1969)

    Google Scholar 

  34. Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  35. Soulignac, F.J.: Bounded, minimal, and short representations of unit interval and unit circular-arc graphs. CoRR (2014). arXiv:1408.3443

  36. Spinrad, J.P.: Efficient Graph Representations. Field Institute Monographs (2003)

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Acknowledgments

We would like to thank an anonymous reviewer for suggestions which improved understandability of Sect. 2. The first, second and sixth authors are supported by ESF Eurogiga project GraDR as GAČR GIG/11/E023, the first two authors also by GAČR 14-14179S and Charles University as GAUK 196213. The fourth author is supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD). The sixth author is supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports, the project NEXLIZ—CZ.1.07/2.3.00/30.0038, which is co-financed by the European Social Fund and the state budget of the Czech Republic, and ESF EuroGIGA project ComPoSe as F.R.S.-FNRS—EUROGIGA NR 13604.

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Authors and Affiliations

  1. Computer Science Institute, Charles University in Prague, Malostranské náměstí 25, 118 00, Prague, Czech Republic

    Pavel Klavík

  2. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University in Prague, Malostranské náměstí 25, 118 00, Prague, Czech Republic

    Jan Kratochvíl & Ignaz Rutter

  3. School of Information Science, Japan Advanced Institute of Science and Technology, Asahidai 1-1, Nomi, Ishikawa, 923-1292, Japan

    Yota Otachi

  4. Faculty of Informatics, Karlsruhe Institute of Technology, Am Fasanengarten 5, 76128, Karlsruhe, Germany

    Ignaz Rutter

  5. Graduate School of Engineering, Kobe University, Rokkodai 1-1, Nada, Kobe, 657-8501, Japan

    Toshiki Saitoh

  6. Department of Mathematics and European Centre of Excellence NTIS (New Technologies for the Information Society), University of West Bohemia, Univerzitní 22, 306 14, Plzeň, Czech Republic

    Maria Saumell

  7. Department of Computer Science, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Tomáš Vyskočil

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  1. Pavel Klavík
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  2. Jan Kratochvíl
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  7. Tomáš Vyskočil
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Correspondence to Pavel Klavík.

Additional information

The conference version of this paper appeared in SWAT 2014 [23].

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Klavík, P., Kratochvíl, J., Otachi, Y. et al. Extending Partial Representations of Proper and Unit Interval Graphs. Algorithmica 77, 1071–1104 (2017). https://doi.org/10.1007/s00453-016-0133-z

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