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Contraction-Bidimensionality of Geometric Intersection Graphs

Authors Julien Baste, Dimitrios M. Thilikos

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Keywords
  • Grid exlusion theorem
  • Bidimensionality
  • Geometric intersection graphs
  • String Graphs

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Abstract

Given a graph G, we define bcg(G) as the minimum k for which G can be contracted to the uniformly triangulated grid Gamma_k. A graph class G has the SQGC property if every graph G in G has treewidth O(bcg(G)c) for some 1 <= c < 2. The SQGC property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a general family of graph classes that satisfy the SQGC property and includes bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for several intersection graph classes of 2-dimensional geometrical objects.

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Julien Baste and Dimitrios M. Thilikos. Contraction-Bidimensionality of Geometric Intersection Graphs. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.IPEC.2017.5

Author Details

Julien Baste
Dimitrios M. Thilikos

References

  1. Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. Journal of Algorithms, 12:308-340, 1991. Google Scholar
  2. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Information and Computation, 243:86-111, 2015. Google Scholar
  3. Julia Chuzhoy. Improved bounds for the excluded grid theorem. CoRR, abs/1602.02629, 2016. Google Scholar
  4. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. Google Scholar
  5. Bruno Courcelle. The expression of graph properties and graph transformations in monadic second-order logic. Handbook of Graph Grammars, pages 313-400, 1997. Google Scholar
  6. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Proceedings of the IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS 2011), pages 150-159, 2011. Google Scholar
  7. Erik D. Demaine. Algorithmic Graph Minors and Bidimensionality. In Proceedings of the 36th International Conference on Graph-theoretic Concepts in Computer Science (WG 2010), pages 2-2, 2010. Google Scholar
  8. Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos. Bidimensional parameters and local treewidth. SIAM Journal on Discrete Mathematics, 18(3):501-511, 2005. Google Scholar
  9. Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. Journal of the ACM, 52(6):866-893, 2005. Google Scholar
  10. Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos. Bidimensional structures: Algorithms, combinatorics and logic. Dagstuhl Reports, 3(3):51-74, 2013. Google Scholar
  11. Erik D. Demaine and MohammadTaghi Hajiaghayi. Fast algorithms for hard graph problems: Bidimensionality, minors, and local treewidth. In Proceedings of the 12th International Symposium on Graph Drawing (GD 2004), volume 3383 of LNCS, pages 517-533, 2004. Google Scholar
  12. Erik D. Demaine and MohammadTaghi Hajiaghayi. Bidimensionality: New connections between FPT algorithms and PTASs. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), pages 590-601, 2005. Google Scholar
  13. Erik D. Demaine and MohammadTaghi Hajiaghayi. The bidimensionality theory and its algorithmic applications. The Computer Journal, 51(3):292-302, 2008. Google Scholar
  14. Erik D. Demaine and MohammadTaghi Hajiaghayi. Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica, 28(1):19-36, 2008. Google Scholar
  15. Erik D. Demaine, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos. The bidimensional theory of bounded-genus graphs. SIAM Journal on Discrete Mathematics, 20(2):357-371, 2006. Google Scholar
  16. Erik D. Demaine, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos. The bidimensional theory of bounded-genus graphs. SIAM Journal on Discrete Mathematics, 20(2):357-371, 2006. Google Scholar
  17. Reinhard Diestel, Tommy R. Jensen, Konstantin Yu. Gorbunov, and Carsten Thomassen. Highly connected sets and the excluded grid theorem. Journal of Combinatorial Theory. Series B, 75(1):61-73, 1999. Google Scholar
  18. Frederic Dorn, Fedor V. Fomin, and Dimitrios M. Thilikos. Fast subexponential algorithm for non-local problems on graphs of bounded genus. In Proceedings of the 10th Scandinavian Workshop on Algorithm Theory (SWAT 2006), volume 4059 of LNCS, pages 172-183, 2006. Google Scholar
  19. Frederic Dorn, Fedor V. Fomin, and Dimitrios M. Thilikos. Catalan structures and dynamic programming in H-minor-free graphs. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pages 631-640, 2008. Google Scholar
  20. Fedor V. Fomin, Erik D. Demaine, and MohammadTaghi Hajiaghayi. Bidimensionality. In Encyclopedia of Algorithms. Springer, 2015. Google Scholar
  21. Fedor V. Fomin, Petr A. Golovach, and Dimitrios M. Thilikos. Contraction obstructions for treewidth. Journal of Combinatorial Theory. Series B, 101(5):302-314, 2011. Google Scholar
  22. Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Bidimensionality and EPTAS. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2011), pages 748-759, 2011. Google Scholar
  23. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Bidimensionality and geometric graphs. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2012), pages 1563-1575, 2012. Google Scholar
  24. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Efficient computation of representative sets with applications in parameterized and exact algorithms. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2014), pages 142-151, 2014. Google Scholar
  25. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and kernels. In Proceedings of the 21th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pages 503-510, 2010. Google Scholar
  26. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and kernels. CoRR, abs/1606.05689, 2016. Google Scholar
  27. Fedor V. Fomin and Dimitrios M. Thilikos Petr Golovach and. Contraction bidimensionality: the accurate picture. In 17th Annual European Symposium on Algorithms, volume 5757 of LNCS, pages 706-717. Springer, 2009. Google Scholar
  28. Fanica Gavril. The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory, Series B, 16(1):47-56, 1974. Google Scholar
  29. Archontia C. Giannopoulou and Dimitrios M. Thilikos. Optimizing the graph minors weak structure theorem. SIAM Journal on Discrete Mathematics, 27(3):1209-1227, 2013. Google Scholar
  30. Alexander Grigoriev, Athanassios Koutsonas, and Dimitrios M. Thilikos. Bidimensionality of Geometric Intersection Graphs. In Proceedings of the 40th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2014), volume 8327 of LNCS, pages 293-305, 2014. Google Scholar
  31. Ken-ichi Kawarabayashi and Yusuke Kobayashi. Linear min-max relation between the treewidth of H-minor-free graphs and its largest grid. In Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012), volume 14 of LIPIcs, pages 278-289, 2012. Google Scholar
  32. Ken-ichi Kawarabayashi, Robin Thomas, and Paul Wollan. New proof of the flat wall theorem. Journal of Combinatorial Theory, Series B, 2017. To appear. Google Scholar
  33. Alexander Leaf and Paul D. Seymour. Tree-width and planar minors. Journal of Combinatorial Theory. Series B, 111:38-53, 2015. Google Scholar
  34. Jiří Matoušek. String graphs and separators. In Geometry, Structure and Randomness in Combinatorics, pages 61-97. Springer, 2014. Google Scholar
  35. Michal Pilipczuk. Problems parameterized by treewidth tractable in single exponential time: A logical approach. In Proceedings of the 36th International Conference on Mathematical Foundations of Computer Science (MFCS 2011), pages 520-531, 2011. Google Scholar
  36. Neil Robertson and Paul D. Seymour. Graph Minors. II. Algorithmic aspects of tree-width. Journal of Algorithms, 7:309-322, 1986. Google Scholar
  37. Neil Robertson and Paul D. Seymour. Graph minors. V. Excluding a planar graph. Journal of Combinatorial Theory. Series B, 41(1):92-114, 1986. Google Scholar
  38. Neil Robertson, Paul D. Seymour, and Robin Thomas. Quickly excluding a planar graph. Journal of Combinatorial Theory. Series B, 62(2):323-348, 1994. Google Scholar
  39. Juanjo Rué, Ignasi Sau, and Dimitrios M. Thilikos. Dynamic programming for H-minor-free graphs. In Proceedings of Computing and Combinatorics - 18th Annual International Conference, (COCOON 2012), pages 86-97, 2012. Google Scholar
  40. Juanjo Rué, Ignasi Sau, and Dimitrios M. Thilikos. Dynamic programming for graphs on surfaces. ACM Transactions on Algorithms, 10(2):1-8, 2014. Google Scholar
  41. Dimitrios M. Thilikos. Graph minors and parameterized algorithm design. In The Multivariate Algorithmic Revolution and Beyond - Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, pages 228-256, 2012. Google Scholar
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