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Peeling and Nibbling the Cactus: Subexponential-Time Algorithms for Counting Triangulations and Related Problems

Authors Dániel Marx, Tillmann Miltzow

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  • computational geometry
  • triangulations
  • exponential-time algorithms

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Abstract

Given a set of n points S in the plane, a triangulation T of S is a maximal set of non-crossing segments with endpoints in S. We present an algorithm that computes the number of triangulations on a given set of n points in time n^{ (11+ o(1)) sqrt{n} }, significantly improving the previous best running time of O(2^n n^2) by Alvarez and Seidel [SoCG 2013]. Our main tool is identifying separators of size O(sqrt{n}) of a triangulation in a canonical way. The definition of the separators are based on the decomposition of the triangulation into nested layers ("cactus graphs"). Based on the above algorithm, we develop a simple and formal framework to count other non-crossing straight-line graphs in n^{O(sqrt{n})} time. We demonstrate the usefulness of the framework by applying it to counting non-crossing Hamilton cycles, spanning trees, perfect matchings, 3-colorable triangulations, connected graphs, cycle decompositions, quadrangulations, 3-regular graphs, and more.

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Dániel Marx and Tillmann Miltzow. Peeling and Nibbling the Cactus: Subexponential-Time Algorithms for Counting Triangulations and Related Problems. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 52:1-52:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SoCG.2016.52

Author Details

Dániel Marx
Tillmann Miltzow

References

  1. Oswin Aichholzer. The path of a triangulation. In SoCG'99, pages 14-23. ACM, 1999. Google Scholar
  2. Victor Alvarez, Karl Bringmann, Radu Curticapean, and Saurabh Ray. Counting triangulations and other crossing-free structures via onion layers. D&C, 53(4):675-690, 2015. URL: http://dx.doi.org/10.1007/s00454-015-9672-3.
  3. Victor Alvarez, Karl Bringmann, and Saurabh Ray. A simple sweep line algorithm for counting triangulations and pseudo-triangulations. CoRR, abs/1312.3188, 2013. URL: http://arxiv.org/abs/1312.3188.
  4. Victor Alvarez, Karl Bringmann, Saurabh Ray, and Raimund Seidel. Counting triangulations and other crossing-free structures approximately. Comput. Geom., 48(5):386-397, 2015. URL: http://dx.doi.org/10.1016/j.comgeo.2014.12.006.
  5. Victor Alvarez and Raimund Seidel. A simple aggregative algorithm for counting triangulations of planar point sets and related problems. In SoCG'13, pages 1-8, 2013. URL: http://dx.doi.org/10.1145/2462356.2462392.
  6. Efthymios Anagnostou and Derek Corneil. Polynomial-time instances of the minimum weight triangulation problem. Computational Geometry, 3(5):247-259, 1993. Google Scholar
  7. David Avis and Komei Fukuda. Reverse search for enumeration. Discrete Applied Mathematics, 65(1):21-46, 1996. Google Scholar
  8. Marshall Bern and David Eppstein. Mesh generation and optimal triangulation. Computing in Euclidean geometry, 1:23-90, 1992. Google Scholar
  9. B. Chazelle. Triangulating a simple polygon in linear time. D&C, 6:485-524, 1991. URL: http://dx.doi.org/10.1007/BF02574703.
  10. Rajesh Hemant Chitnis, MohammadTaghi Hajiaghayi, and Dániel Marx. Tight bounds for Planar Strongly Connected Steiner Subgraph with fixed number of terminals (and extensions). In SODA, pages 1782-1801, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.129.
  11. Mark De Berg, Marc Van Kreveld, Mark Overmars, and Otfried Cheong Schwarzkopf. Computational geometry. Springer, 2000. Google Scholar
  12. Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Bidimensional parameters and local treewidth. SIAM J. Discrete Math., 18(3):501-511, 2004. URL: http://dx.doi.org/10.1137/S0895480103433410.
  13. Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Fixed-parameter algorithms for (k,r)-Center in planar graphs and map graphs. ACM Transactions on Algorithms, 1(1):33-47, 2005. URL: http://dx.doi.org/10.1145/1077464.1077468.
  14. Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM, 52(6):866-893, 2005. URL: http://dx.doi.org/10.1145/1101821.1101823.
  15. Erik D. Demaine and Mohammad Taghi Hajiaghayi. Fast algorithms for hard graph problems: Bidimensionality, minors, and local treewidth. In Graph Drawing, pages 517-533, 2004. URL: http://dx.doi.org/10.1007/978-3-540-31843-9_57.
  16. Erik D. Demaine and MohammadTaghi Hajiaghayi. The bidimensionality theory and its algorithmic applications. Comput. J., 51(3):292-302, 2008. URL: http://dx.doi.org/10.1093/comjnl/bxm033.
  17. Erik D. Demaine and MohammadTaghi Hajiaghayi. Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica, 28(1):19-36, 2008. URL: http://dx.doi.org/10.1007/s00493-008-2140-4.
  18. Frederic Dorn, Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs. In STACS, pages 251-262, 2010. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2010.2459.
  19. Frederic Dorn, Fedor V. Fomin, and Dimitrios M. Thilikos. Subexponential parameterized algorithms. Computer Science Review, 2(1):29-39, 2008. URL: http://dx.doi.org/10.1016/j.cosrev.2008.02.004.
  20. Frederic Dorn, Eelko Penninkx, Hans L. Bodlaender, and Fedor V. Fomin. Efficient exact algorithms on planar graphs: Exploiting sphere cut decompositions. Algorithmica, 58(3):790-810, 2010. URL: http://dx.doi.org/10.1007/s00453-009-9296-1.
  21. Adrian Dumitrescu, Bernd Gärtner, Samuele Pedroni, and Emo Welzl. Enumerating triangulation paths. Computational Geometry, 20(1):3-12, 2001. Google Scholar
  22. Herbert Edelsbrunner and John Harer. Computational topology: an introduction. American Mathematical Soc., 2010. Google Scholar
  23. Steve Fisk. A short proof of chvátal’s watchman theorem. J. Comb. Theory, 24(3):374, 1978. URL: http://dx.doi.org/10.1016/0095-8956(78)90059-X.
  24. Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Subexponential algorithms for partial cover problems. Inf. Process. Lett., 111(16):814-818, 2011. URL: http://dx.doi.org/10.1016/j.ipl.2011.05.016.
  25. Fedor V. Fomin and Dimitrios M. Thilikos. Dominating sets in planar graphs: Branch-width and exponential speed-up. SIAM J. Comput., 36(2):281-309, 2006. URL: http://dx.doi.org/10.1137/S0097539702419649.
  26. Peter D Gilbert. New results on planar triangulations. Technical report, 1979. Google Scholar
  27. Leonidas J. Guibas, John Hershberger, Daniel Leven, Micha Sharir, and Robert Endre Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2:209-233, 1987. URL: http://dx.doi.org/10.1007/BF01840360.
  28. John Hershberger and Subhash Suri. A pedestrian approach to ray shooting: Shoot a ray, take a walk. Journal of Algorithms, 18(3):403-431, 1995. Google Scholar
  29. Øyvind Hjelle and Morten Dæhlen. Triangulations and Applications. Springer, 2006. Google Scholar
  30. Marek Karpinski, Andrzej Lingas, and Dzmitry Sledneu. A QPTAS for the base of the number of crossing-free structures on a planar point set. In ICALP'15, pages 785-796, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_64.
  31. Philip N. Klein and Dániel Marx. Solving Planar k-Terminal Cut in O(n^c√k) time. In ICALP (1), pages 569-580, 2012. URL: http://dx.doi.org/10.1007/978-3-642-31594-7_48.
  32. Philip N. Klein and Dániel Marx. A subexponential parameterized algorithm for Subset TSP on planar graphs. In SODA, pages 1812-1830, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.131.
  33. GT Klincsek. Minimal triangulations of polygonal domains. Ann. Discrete Math, 9:121-123, 1980. Google Scholar
  34. Christian Knauer and Andreas Spillner. A fixed-parameter algorithm for the minimum weight triangulation problem based on small graph separators. In Fedor V. Fomin, editor, Graph-Theoretic Concepts in Computer Science, 32nd International Workshop, WG 2006, Bergen, Norway, June 22-24, 2006, Revised Papers, volume 4271 of Lecture Notes in Computer Science, pages 49-57. Springer, 2006. URL: http://dx.doi.org/10.1007/11917496_5.
  35. Andrzej Lingas. Subexponential-time algorithms for minimum weight triangulations and related problems. In CCCG'98, 1998. URL: http://cgm.cs.mcgill.ca/cccg98/proceedings/cccg98-lingas-subexponential.ps.gz.
  36. Richard J Lipton and Robert Endre Tarjan. A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 36(2):177-189, 1979. Google Scholar
  37. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the Exponential Time Hypothesis. Bulletin of the EATCS, 105:41-72, 2011. Google Scholar
  38. Dániel Marx and Tillmann Miltzow. Peeling and nibbling the cactus: Subexponential-time algorithms for counting triangulations and related problems. CoRR, to be announced, 2016. Google Scholar
  39. Dániel Marx and Michal Pilipczuk. Optimal parameterized algorithms for planar facility location problems using voronoi diagrams. In ESA'15, pages 865-877. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_72.
  40. Marcin Pilipczuk, Michal Pilipczuk, Piotr Sankowski, and Erik Jan van Leeuwen. Network sparsification for steiner problems on planar and bounded-genus graphs. In FOCS'14, pages 276-285. URL: http://dx.doi.org/10.1109/FOCS.2014.37.
  41. Marcin Pilipczuk, Michał Pilipczuk, Piotr Sankowski, and Erik Jan van Leeuwen. Subexponential-time parameterized algorithm for Steiner Tree on planar graphs. In STACS, pages 353-364, 2013. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2013.353.
  42. S Ray and R Seidel. A simple and less slow method for counting triangulations and for related problems. In EuroCG'04, pages 77-80, 2004. Google Scholar
  43. Micha Sharir and Adam Sheffer. Counting triangulations of planar point sets. Electr. J. Comb., 18(1), 2011. URL: http://www.combinatorics.org/Volume_18/Abstracts/v18i1p70.html.
  44. Dimitrios M. Thilikos. Fast sub-exponential algorithms and compactness in planar graphs. In ESA, pages 358-369, 2011. URL: http://dx.doi.org/10.1007/978-3-642-23719-5_31.
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