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Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-Wideness

Authors Wojciech Nadara, Marcin Pilipczuk, Roman Rabinovich, Felix Reidl, Sebastian Siebertz



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Wojciech Nadara
  • Institute of Informatics, University of Warsaw, Poland
Marcin Pilipczuk
  • Institute of Informatics, University of Warsaw, Poland
Roman Rabinovich
  • Lehrstuhl für Logic und Semantik, Technische Universität Berlin, Germany
Felix Reidl
  • Department of Computer Science, Royal Holloway University of London, UK
Sebastian Siebertz
  • Institute of Informatics, University of Warsaw, Poland

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Wojciech Nadara, Marcin Pilipczuk, Roman Rabinovich, Felix Reidl, and Sebastian Siebertz. Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-Wideness. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 14:1-14:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SEA.2018.14

Abstract

The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we study two structural properties of these graph classes that are of particular importance in this context, namely the property of having bounded generalized coloring numbers and the property of being uniformly quasi-wide. We provide experimental evaluations of several algorithms that approximate these parameters on real-world graphs. On the theoretical side, we provide a new algorithm for uniform quasi-wideness with polynomial size guarantees in graph classes of bounded expansion and show a lower bound indicating that the guarantees of this algorithm are close to optimal in graph classes with fixed excluded minor.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Empirical Evaluation of Algorithms
  • Sparse Graph Classes
  • Generalized Coloring Numbers
  • Uniform Quasi-Wideness

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References

  1. Gephi datasets. URL: https://github.com/gephi/gephi/wiki/Datasets.
  2. LEDA. URL: http://www.algorithmic-solutions.com/leda/index.htm.
  3. Recent trends in kernelization: theory and experimental evaluation - project website, 2018. URL: http://kernelization-experiments.mimuw.edu.pl.
  4. Lada A Adamic and Natalie Glance. The political blogosphere and the 2004 US election: divided they blog. In Proceedings of the 3rd International Workshop on Link Discovery, pages 36-43. ACM, 2005. Google Scholar
  5. Jochen Alber, Michael R. Fellows, and Rolf Niedermeier. Polynomial-time data reduction for dominating set. Journal of the ACM, 51(3):363-384, 2004. URL: http://dx.doi.org/10.1145/990308.990309.
  6. Saeed Akhoondian Amiri, Patrice Ossona de Mendez, Roman Rabinovich, and Sebastian Siebertz. Distributed domination on graph classes of bounded expansion. CoRR, abs/1702.02848, 2017. Google Scholar
  7. Vladimir Batagelj and Andrej Mrvar. Pajek datasets. http://vlado.fmf.uni-lj.si/pub/networks/data/, 2006.
  8. Hans L. Bodlaender. Treewidth: Algorithmic techniques and results. In International Symposium on Mathematical Foundations of Computer Science, pages 19-36. Springer, 1997. Google Scholar
  9. Hans L. Bodlaender and Arie M. C. A. Koster. Treewidth computations I. Upper bounds. Information and Computation, 208(3):259-275, 2010. URL: http://dx.doi.org/10.1016/j.ic.2009.03.008.
  10. Dongbo Bu, Yi Zhao, Lun Cai, Hong Xue, Xiaopeng Zhu, Hongchao Lu, Jingfen Zhang, Shiwei Sun, Lunjiang Ling, Nan Zhang, et al. Topological structure analysis of the protein-protein interaction network in budding yeast. Nucleic acids research, 31(9):2443-2450, 2003. Google Scholar
  11. Eunjoon Cho, Seth A. Myers, and Jure Leskovec. Friendship and mobility: user movement in location-based social networks. In Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Diego, CA, USA, August 21-24, 2011, pages 1082-1090, 2011. URL: http://dx.doi.org/10.1145/2020408.2020579.
  12. Fan Chung and Linyuan Lu. The average distances in random graphs with given expected degrees. Proceedings of the National Academy of Sciences, 99(25):15879-15882, 2002. Google Scholar
  13. Fan Chung and Linyuan Lu. Connected components in random graphs with given expected degree sequences. Annals of combinatorics, 6(2):125-145, 2002. Google Scholar
  14. Fan R. K. Chung and Linyuan Lu. The average distance in a random graph with given expected degrees. Internet Mathematics, 1(1):91-113, 2003. URL: http://dx.doi.org/10.1080/15427951.2004.10129081.
  15. Anuj Dawar. Homomorphism preservation on quasi-wide classes. Journal of Computer and System Sciences, 76(5):324-332, 2010. Google Scholar
  16. Anuj Dawar and Stephan Kreutzer. Domination problems in nowhere-dense classes. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS, pages 157-168, 2009. Google Scholar
  17. Holger Dell, Thore Husfeldt, Bart M. P. Jansen, Petteri Kaski, Christian Komusiewicz, and Frances A. Rosamond. The First Parameterized Algorithms and Computational Experiments Challenge. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016), volume 63 of Leibniz International Proceedings in Informatics (LIPIcs), pages 30:1-30:9, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.IPEC.2016.30.
  18. Erik D. Demaine and MohammadTaghi Hajiaghayi. The bidimensionality theory and its algorithmic applications. The Computer Journal, 51(3):292-302, 2007. Google Scholar
  19. Erik D. Demaine, MohammadTaghi Hajiaghayi, and Ken-ichi Kawarabayashi. Algorithmic graph minor theory: Improved grid minor bounds and wagner’s contraction. Algorithmica, 54(2):142-180, 2009. Google Scholar
  20. Erik D. Demaine, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, Somnath Sikdar, and Blair D. Sullivan. Structural sparsity of complex networks: Random graph models and linear algorithms. CoRR, abs/1406.2587, 2014. URL: http://arxiv.org/abs/1406.2587.
  21. Pål Grønås Drange, Markus Sortland Dregi, Fedor V. Fomin, Stephan Kreutzer, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, Felix Reidl, Fernando Sánchez Villaamil, Saket Saurabh, Sebastian Siebertz, and Somnath Sikdar. Kernelization and sparseness: the case of dominating set. In 33rd Symposium on Theoretical Aspects of Computer Science, STACS, pages 31:1-31:14, 2016. Google Scholar
  22. Zdenek Dvorak. Constant-factor approximation of the domination number in sparse graphs. European Journal of Combinatorics, 34(5):833-840, 2013. Google Scholar
  23. Zdenek Dvorak, Daniel Král, and Robin Thomas. Testing first-order properties for subclasses of sparse graphs. Journal of the ACM, 60(5):36:1-36:24, 2013. Google Scholar
  24. Kord Eickmeyer, Archontia C. Giannopoulou, Stephan Kreutzer, O-joung Kwon, Michał Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. Neighborhood complexity and kernelization for nowhere dense classes of graphs. In 44th International Colloquium on Automata, Languages, and Programming, ICALP, pages 63:1-63:14, 2017. Google Scholar
  25. Jakub Gajarský, Petr Hliněný, Jan Obdržálek, Sebastian Ordyniak, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, and Somnath Sikdar. Kernelization using structural parameters on sparse graph classes. Journal of Computer and System Sciences, 84:219-242, 2017. Google Scholar
  26. Michelle Girvan and Mark E. J. Newman. Community structure in social and biological networks. Proceedings of the National Academy of Sciences, 99(12):7821-7826, 2002. Google Scholar
  27. Kwang-Il Goh, Michael E. Cusick, David Valle, Barton Childs, Marc Vidal, and Albert-László Barabási. The human disease network. Proceedings of the National Academy of Sciences, 104(21):8685-8690, 2007. Google Scholar
  28. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. Journal of the ACM, 64(3):17:1-17:32, 2017. Google Scholar
  29. Paul W. Holland, Kathryn B. Laskey, and Samuel Leinhardt. Stochastic blockmodels: First steps. Social networks, 5(2):109-137, 1983. Google Scholar
  30. Wojciech Kazana and Luc Segoufin. Enumeration of first-order queries on classes of structures with bounded expansion. In Proceedings of the 32nd ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS, pages 297-308, 2013. Google Scholar
  31. Henry A. Kierstead and Daqing Yang. Orderings on graphs and game coloring number. Order, 20:255-264, 2003. Google Scholar
  32. Bryan Klimt and Yiming Yang. Introducing the Enron corpus. In CEAS, 2004. Google Scholar
  33. Donald Ervin Knuth. The Stanford GraphBase: a platform for combinatorial computing, volume 37. Addison-Wesley Reading, 1993. Google Scholar
  34. Stephan Kreutzer, Michał Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. The generalised colouring numbers on classes of bounded expansion. In 41st International Symposium on Mathematical Foundations of Computer Science, MFCS, 2016. Google Scholar
  35. Stephan Kreutzer, Roman Rabinovich, and Sebastian Siebertz. Polynomial kernels and wideness properties of nowhere dense graph classes. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1533-1545, 2017. Google Scholar
  36. Jérôme Kunegis. KONECT - the Koblenz network collection. In Proc. Int. Web Observatory Workshop, pages 1343-1350, 2013. URL: http://konect.uni-koblenz.de.
  37. Jure Leskovec, Daniel Huttenlocher, and Jon Kleinberg. Signed networks in social media. In Proceedings of the SIGCHI conference on human factors in computing systems, pages 1361-1370. ACM, 2010. Google Scholar
  38. Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. Graphs over time: densification laws, shrinking diameters and possible explanations. In Proceedings of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining, pages 177-187. ACM, 2005. Google Scholar
  39. Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network dataset collection. http://snap.stanford.edu/data, 2014.
  40. David Lusseau, Karsten Schneider, Oliver J. Boisseau, Patti Haase, Elisabeth Slooten, and Steve M. Dawson. The bottlenose dolphin community of doubtful sound features a large proportion of long-lasting associations. Behavioral Ecology and Sociobiology, 54(4):396-405, 2003. Google Scholar
  41. Wojciech Nadara, Marcin Pilipczuk, Felix Reidl, Roman Rabinovich, and Sebastian Siebertz. Empirical evaluation of approximation algorithms for generalized graph coloring and uniform quasi-wideness. code repository, 2018. https://bitbucket.org/marcin_pilipczuk/wcol-uqw-experiments. Google Scholar
  42. Jaroslav Nešetřil and Patrice Ossona de Mendez. Grad and classes with bounded expansion I. decompositions. European Journal of Combinatorics, 29(3):760-776, 2008. Google Scholar
  43. Jaroslav Nešetřil and Patrice Ossona de Mendez. Grad and classes with bounded expansion II. algorithmic aspects. Europeean Journal of Combinatorics, 29(3):777-791, 2008. Google Scholar
  44. Jaroslav Nešetřil and Patrice Ossona de Mendez. Grad and classes with bounded expansion III. restricted graph homomorphism dualities. European Journal of Combinatorics, 29(4):1012-1024, 2008. Google Scholar
  45. Jaroslav Nešetřil and Patrice Ossona de Mendez. First order properties on nowhere dense structures. The Journal of Symbolic Logic, 75(3):868-887, 2010. Google Scholar
  46. Jaroslav Nešetřil and Patrice Ossona de Mendez. On nowhere dense graphs. European Journal of Combinatorics, 32(4):600-617, 2011. Google Scholar
  47. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. Google Scholar
  48. Mark EJ Newman. The structure of scientific collaboration networks. Proceedings of the national academy of sciences, 98(2):404-409, 2001. Google Scholar
  49. Mark EJ Newman. Finding community structure in networks using the eigenvectors of matrices. Physical review E, 74(3):036104, 2006. Google Scholar
  50. Michael P. O'Brien and Blair D. Sullivan. Experimental evaluation of counting subgraph isomorphisms in classes of bounded expansion. CoRR, abs/1712.06690, 2017. URL: http://arxiv.org/abs/1712.06690.
  51. Tobias Oelschlägel. Treewidth from Treedepth. Bachelor’s thesis, RWTH Aachen University, Germany, 2014. Google Scholar
  52. Michał Pilipczuk, Sebastian Siebertz, and Szymon Toruńczyk. On the number of types in sparse graphs. arXiv preprint arXiv:1705.09336, 2017. Google Scholar
  53. Felix Reidl. Structural sparseness and complex networks. Dr., Aachen, Techn. Hochsch., Aachen, 2016. Aachen, Techn. Hochsch., Diss., 2015. URL: http://publications.rwth-aachen.de/record/565064.
  54. Felix Reidl, Fernando Sánchez Villaamil, and Konstantinos Stavropoulos. Characterising bounded expansion by neighbourhood complexity. CoRR, abs/1603.09532, 2016. Google Scholar
  55. Matthew Richardson, Rakesh Agrawal, and Pedro Domingos. Trust management for the semantic web. In International semantic Web conference, pages 351-368. Springer, 2003. Google Scholar
  56. Matei Ripeanu, Ian Foster, and Adriana Iamnitchi. Mapping the Gnutella Network. IEEE Internet Computing, 6(1):50-57, 2002. Google Scholar
  57. Neil Robertson and Paul D. Seymour. Graph minors I-XXII, 1982-2010. Google Scholar
  58. Ryan A. Rossi and Nesreen K. Ahmed. The network data repository with interactive graph analytics and visualization. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, 2015. URL: http://networkrepository.com.
  59. Fernando Sanchez Villaamil. About Treedepth and Related Notions. Dissertation, RWTH Aachen University, Aachen, 2017. URL: http://dx.doi.org/10.18154/RWTH-2017-09829.
  60. Sebastian Siebertz. Reconfiguration on nowhere dense graph classes. CoRR, abs/1707.06775, 2017. Google Scholar
  61. Juliette Stehlé, N. Voirin, Alain Barrat, Ciro Cattuto, Lorenzo Isella, Jean-François Pinton, Marco Quaggiotto, Wouter Van den Broeck, C. Régis, B. Lina, and P. Vanhems. High-resolution measurements of face-to-face contact patterns in a primary school. PLOS ONE, 6(8):e23176, 08 2011. Google Scholar
  62. Jan van den Heuvel, Patrice Ossona de Mendez, Daniel A. Quiroz, Roman Rabinovich, and Sebastian Siebertz. On the generalised colouring numbers of graphs that exclude a fixed minor. European Journal of Combinatorics, 66:129-144, 2017. Google Scholar
  63. Jan van den Heuvel, Stephan Kreutzer, Michał Pilipczuk, Daniel A. Quiroz, Roman Rabinovich, and Sebastian Siebertz. Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion. In 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS, pages 1-11, 2017. Google Scholar
  64. Duncan J Watts and Steven H Strogatz. Collective dynamics of `small-world' networks. nature, 393(6684):440, 1998. Google Scholar
  65. John G. White, Eileen Southgate, J. Nichol Thomson, and Sydney Brenner. The structure of the nervous system of the nematode caenorhabditis elegans: the mind of a worm. Phil. Trans. R. Soc. Lond, 314:1-340, 1986. Google Scholar
  66. Wayne W. Zachary. An information flow model for conflict and fission in small groups. Journal of anthropological research, pages 452-473, 1977. Google Scholar
  67. Xuding Zhu. Colouring graphs with bounded generalized colouring number. Discrete Math., 309:5562-5568, 2009. Google Scholar
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