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Algorithmic Properties of Sparse Digraphs

Authors Stephan Kreutzer, Irene Muzi, Patrice Ossona de Mendez, Roman Rabinovich, Sebastian Siebertz



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Stephan Kreutzer
  • Technische Universität Berlin, Germany
Irene Muzi
  • University of Warsaw, Poland
Patrice Ossona de Mendez
  • Centre d'Analyse et de Mathématiques Sociales (CNRS, UMR 8557), Paris, France,
  • Computer Science Institute of Charles University (IUUK), Prague, Czech Republic
Roman Rabinovich
  • Technische Universität Berlin, Germany
Sebastian Siebertz
  • Humboldt-Universität zu Berlin, Germany

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Stephan Kreutzer, Irene Muzi, Patrice Ossona de Mendez, Roman Rabinovich, and Sebastian Siebertz. Algorithmic Properties of Sparse Digraphs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 46:1-46:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.STACS.2019.46

Abstract

The notions of bounded expansion [Nešetřil and Ossona de Mendez, 2008] and nowhere denseness [Nešetřil and Ossona de Mendez, 2011], introduced by Nešetřil and Ossona de Mendez as structural measures for undirected graphs, have been applied very successfully in algorithmic graph theory. We study the corresponding notions of directed bounded expansion and nowhere crownfulness on directed graphs, introduced by Kreutzer and Tazari [Kreutzer and Tazari, 2012]. The classes of directed graphs having those properties are very general classes of sparse directed graphs, as they include, on one hand, all classes of directed graphs whose underlying undirected class has bounded expansion, such as planar, bounded-genus, and H-minor-free graphs, and on the other hand, they also contain classes whose underlying undirected class is not even nowhere dense. We show that many of the algorithmic tools that were developed for undirected bounded expansion classes can, with some care, also be applied in their directed counterparts, and thereby we highlight a rich algorithmic structure theory of directed bounded expansion and nowhere crownful classes.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Fixed parameter tractability
Keywords
  • Directed graphs
  • graph algorithms
  • parameterized complexity
  • approximation

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References

  1. Hans Adler and Isolde Adler. Interpreting nowhere dense graph classes as a classical notion of model theory. European Journal of Combinatorics, 36:322-330, 2014. Google Scholar
  2. Jochen Alber, Michael R Fellows, and Rolf Niedermeier. Polynomial-time data reduction for dominating set. Journal of the ACM (JACM), 51(3):363-384, 2004. Google Scholar
  3. Saeed Akhoondian Amiri, Patrice Ossona de Mendez, Roman Rabinovich, and Sebastian Siebertz. Distributed Domination on Graph Classes of Bounded Expansion. In Proceedings of the 30th on Symposium on Parallelism in Algorithms and Architectures, SPAA, pages 143-151, 2018. Google Scholar
  4. Brenda S. Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM (JACM), 41(1):153-180, 1994. Google Scholar
  5. Jørgen Bang-Jensen and Gregory Z Gutin. Digraphs: theory, algorithms and applications. Springer Science &Business Media, 2008. Google Scholar
  6. János Barát. Directed path-width and monotonicity in digraph searching. Graphs and Combinatorics, 22(2):161-172, 2006. Google Scholar
  7. Dietmar Berwanger, Anuj Dawar, Paul Hunter, and Stephan Kreutzer. DAG-width and parity games. In Annual Symposium on Theoretical Aspects of Computer Science, pages 524-536. Springer, 2006. Google Scholar
  8. Hans L Bodlaender, Fedor V Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M Thilikos. (Meta) kernelization. In Foundations of Computer Science, 2009. FOCS'09. 50th Annual IEEE Symposium on, pages 629-638. IEEE, 2009. Google Scholar
  9. Marthe Bonamy, Łukasz Kowalik, Michał Pilipczuk, and Arkadiusz Socała. Linear kernels for outbranching problems in sparse digraphs. Algorithmica, pages 1-30, 2015. Google Scholar
  10. Hervé Brönnimann and Michael T Goodrich. Almost optimal set covers in finite VC-dimension. Discrete &Computational Geometry, 14(4):463-479, 1995. Google Scholar
  11. Julia Chuzhoy and Shi Li. A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2. J. ACM, 63(5):45:1-45:51, November 2016. URL: http://dx.doi.org/10.1145/2893472.
  12. Anuj Dawar, Martin Grohe, Stephan Kreutzer, and Nicole Schweikardt. Approximation schemes for first-order definable optimisation problems. In 21st Annual IEEE Symposium on Logic in Computer Science, 2006, pages 411-420. IEEE, 2006. Google Scholar
  13. 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 2009, December 15-17, 2009, IIT Kanpur, India, pages 157-168, 2009. Google Scholar
  14. E. Demaine and M. Hajiaghayi. The bidimensionality theory and its algorithmic applications. The Computer Journal, pages 332-337, 2008. Google Scholar
  15. E. D. Demaine, M. Hajiaghayi, and K. Kawarabayashi. Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring. In 46th Annual Symposium on Foundations of Computer Science (FOCS), pages 637-646, 2005. Google Scholar
  16. 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 (JACM), 52(6):866-893, 2005. Google Scholar
  17. 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://springerlink.metapress.com/openurl.asp?genre=article&issn=0302-9743&volume=3383&spage=517.
  18. Erik D. Demaine and Mohammad Taghi Hajiaghayi. Bidimensionality: new connections between FPT algorithms and PTASs. In Symp. on Discrete Algorithms (SODA), pages 590-601, 2005. URL: http://dx.doi.org/10.1145/1070432.1070514.
  19. 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,
  20. Pål Grønås Drange, Markus Sortland Dregi, Fedor V. Fomin, Stephan Kreutzer, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, Felix Reidl, Fernando Sánchez Villaamil, Saket Saurabh, Sebastian Siebertz, and Somnath Sikdar. Kernelization and Sparseness: the Case of Dominating Set. In STACS, volume 47 of LIPIcs, pages 31:1-31:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  21. Zdeněk Dvořák. Constant-factor approximation of the domination number in sparse graphs. European Journal of Combinatorics, 34(5):833-840, 2013. Google Scholar
  22. Zdeněk Dvořák. On distance r-dominating and 2r-independent sets in sparse graphs. arXiv preprint arXiv:1710.10010, 2017. Google Scholar
  23. Zdeněk Dvořák, Daniel Král, and Robin Thomas. Testing first-order properties for subclasses of sparse graphs. Journal of the ACM (JACM), 60(5):36, 2013. Google Scholar
  24. Eduard Eiben, Mithilesh Kumar, Amer E. Mouawad, Fahad Panolan, and Sebastian Siebertz. Lossy Kernels for Connected Dominating Set on Sparse Graphs. In 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France, pages 29:1-29:15, 2018. Google Scholar
  25. Kord Eickmeyer, Archontia C Giannopoulou, Stephan Kreutzer, O-joung Kwon, Michał Pilipczuk, Roman Rabinovich, Sebastian Siebertz, et al. Neighborhood complexity and kernelization for nowhere dense classes of graphs. arXiv preprint arXiv:1612.08197, 2016. Google Scholar
  26. Grzegorz Fabiański, Michał Pilipczuk, Sebastian Siebertz, and Szymon Toruńczyk. Progressive Algorithms for Domination and Independence. CoRR, abs/1811.06799, 2018. URL: http://arxiv.org/abs/1811.06799.
  27. 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. Google Scholar
  28. Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Bidimensionality and EPTAS. In SODA, pages 748-759, 2011. URL: http://www.siam.org/proceedings/soda/2011/SODA11_058_fominf.pdf.
  29. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and Kernels. In SODA, pages 503-510, 2010. URL: http://www.siam.org/proceedings/soda/2010/SODA10_043_fominf.pdf.
  30. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M Thilikos. Bidimensionality and kernels. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 503-510. SIAM, 2010. Google Scholar
  31. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M Thilikos. Linear kernels for (connected) dominating set on H-minor-free graphs. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 82-93. Society for Industrial and Applied Mathematics, 2012. Google Scholar
  32. Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M Thilikos. Linear kernels for (connected) dominating set on graphs with excluded topological subgraphs. In 30th International Symposium on Theoretical Aspects of Computer Science, page 92, 2013. Google Scholar
  33. Jakub Gajarský, Petr Hliněný, Jan Obdržálek, Sebastian Ordyniak, Felix Reidl, Peter Rossmanith, Fernando Sanchez Villaamil, and Somnath Sikdar. Kernelization using structural parameters on sparse graph classes. In European Symposium on Algorithms, pages 529-540. Springer, 2013. Google Scholar
  34. Robert Ganian, Petr Hliněný, Joachim Kneis, Alexander Langer, Jan Obdržálek, and Peter Rossmanith. On digraph width measures in parameterized algorithmics. In International Workshop on Parameterized and Exact Computation, pages 185-197. Springer, 2009. Google Scholar
  35. Robert Ganian, Petr Hlinený, Joachim Kneis, Daniel Meister, Jan Obdrzálek, Peter Rossmanith, and Somnath Sikdar. Are there any good digraph width measures? J. Comb. Theory, Ser. B, 116:250-286, 2016. URL: http://dx.doi.org/10.1016/j.jctb.2015.09.001.
  36. Martin Grohe. Local tree-width, excluded minors, and approximation algorithms. Combinatorica, 23(4):613-632, 2003. Google Scholar
  37. Martin Grohe, Stephan Kreutzer, Roman Rabinovich, Sebastian Siebertz, and Konstantinos Stavropoulos. Colouring and covering nowhere dense graphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 325-338. Springer, 2015. Google Scholar
  38. Martin Grohe, Stephan Kreutzer, and Sebastian Siebertz. Deciding first-order properties of nowhere dense graphs. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 89-98. ACM, 2014. Google Scholar
  39. Sariel Har-Peled and Kent Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. SIAM Journal on Computing, 46(6):1712-1744, 2017. Google Scholar
  40. Paul Hunter and Stephan Kreutzer. Digraph measures: Kelly decompositions, games, and orderings. Theoretical Computer Science, 399(3):206-219, 2008. Google Scholar
  41. M. Jones, D. Lokshtanov, M.S. Ramanujan, S. Saurabh, and O. Suchy. Parameterized Complexity of Directed Steiner Tree on Sparse Graphs. In Proceedings of European Symposium on Algorithms (ESA 2013), 2013. Google Scholar
  42. Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Springer, 1972. Google Scholar
  43. 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-SIGAI symposium on Principles of database systems, pages 297-308. ACM, 2013. Google Scholar
  44. Stephan Kreutzer, Irene Muzi, Patrice Ossona de Mendez, Roman Rabinovich, and Sebastian Siebertz. Algorithmic Properties of Sparse Digraphs. CoRR, abs/1707.01701, 2017. Google Scholar
  45. 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, pages 1533-1545. SIAM, 2017. Google Scholar
  46. 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 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1533-1545, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.100.
  47. Stephan Kreutzer, Roman Rabinovich, Sebastian Siebertz, and Grischa Weberstädt. Structural Properties and Constant Factor-Approximation of Strong Distance-r Dominating Sets in Sparse Directed Graphs. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), volume 66 of Leibniz International Proceedings in Informatics (LIPIcs), pages 48:1-48:15, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2017.48.
  48. Stephan Kreutzer and Siamak Tazari. Directed nowhere dense classes of graphs. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 1552-1562. SIAM, 2012. Google Scholar
  49. Daniel Lokshtanov, Amer E Mouawad, Fahad Panolan, MS Ramanujan, and Saket Saurabh. Reconfiguration on sparse graphs. In Workshop on Algorithms and Data Structures, pages 506-517. Springer, 2015. Google Scholar
  50. Maryanthe Malliaris and Saharon Shelah. Regularity lemmas for stable graphs. Transactions of the American Mathematical Society, 366(3):1551-1585, 2014. Google Scholar
  51. N. Misra, G. Philip, V. Raman, S. Saurabh, and S. Sikdar. FPT algorithms for connected feedback vertex set. In WALCOM 2010, pages 269-280, 2010. Google Scholar
  52. Daniel Mölle, Stefan Richter, and Peter Rossmanith. Enumerate and expand: Improved algorithms for connected vertex cover and tree cover. Theory of Computing Systems, 43(2):234-253, 2008. Google Scholar
  53. 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, pages 14:1-14:16, 2018. Google Scholar
  54. J. Nederlof. Fast polynomial-space algorithms using Möbius inversion: Improving on Steiner tree and related problems. In Proc. of ICALP 2009, pages 713-725, 2009. Google Scholar
  55. Jaroslav Nešetřil and Patrice Ossona de Mendez. Linear time low tree-width partitions and algorithmic consequences. In Proceedings of the thirty-eighth annual ACM Symposium on Theory of Computing (STOC), pages 391-400. ACM, 2006. Google Scholar
  56. 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
  57. Jaroslav Nešetřil and Patrice Ossona de Mendez. First order properties on nowhere dense structures. The Journal of Symbolic Logic, 75(03):868-887, 2010. Google Scholar
  58. Jaroslav Nešetřil and Patrice Ossona de Mendez. On nowhere dense graphs. European Journal of Combinatorics, 32(4):600-617, 2011. Google Scholar
  59. Jan Obdržálek. DAG-width: connectivity measure for directed graphs. In Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pages 814-821. Society for Industrial and Applied Mathematics, 2006. Google Scholar
  60. Geevarghese Philip, Venkatesh Raman, and Somnath Sikdar. Polynomial kernels for dominating set in graphs of bounded degeneracy and beyond. ACM Transactions on Algorithms (TALG), 9(1):11, 2012. Google Scholar
  61. Michał Pilipczuk and Sebastian Siebertz. Kernelization and approximation of distance-r independent sets on nowhere dense graphs. arXiv preprint arXiv:1809.05675, 2018. Google Scholar
  62. Michał Pilipczuk, Sebastian Siebertz, and Szymon Torunczyk. On the number of types in sparse graphs. arXiv preprint arXiv:1705.09336, 2017. Google Scholar
  63. Hans Jürgen Prömel and Angelika Steger. The Steiner tree problem: a tour through graphs, algorithms, and complexity. Springer Science &Business Media, 2012. Google Scholar
  64. Ran Raz and Shmuel Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pages 475-484. ACM, 1997. Google Scholar
  65. Felix Reidl, Fernando Sánchez Villaamil, and Konstantinos Stavropoulos. Characterising bounded expansion by neighbourhood complexity. Eur. J. Comb., 75:152-168, 2019. Google Scholar
  66. N. Robertson and P.D. Seymour. Graph minors XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B, 63:65-110, 1995. Google Scholar
  67. Mohammad Ali Safari. D-width: A more natural measure for directed tree width. In International Symposium on Mathematical Foundations of Computer Science, pages 745-756. Springer, 2005. Google Scholar
  68. Norbert Sauer. On the density of families of sets. Journal of Combinatorial Theory, Series A, 13(1):145-147, 1972. Google Scholar
  69. Luc Segoufin and Alexandre Vigny. Constant Delay Enumeration for FO Queries over Databases with Local Bounded Expansion. In Michael Benedikt and Giorgio Orsi, editors, 20th International Conference on Database Theory (ICDT 2017), volume 68 of Leibniz International Proceedings in Informatics (LIPIcs), pages 20:1-20:16, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ICDT.2017.20.
  70. Saharon Shelah. A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific Journal of Mathematics, 41(1):247-261, 1972. Google Scholar
  71. Sebastian Siebertz. Reconfiguration on Nowhere Dense Graph Classes. Electr. J. Comb., 25(3):P3.24, 2018. Google Scholar
  72. VN Vapnik and A Ya Chervonenkis. On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities. Measures of Complexity, 16(2):11, 1971. Google Scholar
  73. Xuding Zhu. Colouring graphs with bounded generalized colouring number. Discrete Mathematics, 309(18):5562-5568, 2009. Google Scholar
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