Turbocharging Heuristics for Weak Coloring Numbers

Authors Alexander Dobler, Manuel Sorge, Anaïs Villedieu

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Alexander Dobler
  • TU Wien, Austria
Manuel Sorge
  • TU Wien, Austria
Anaïs Villedieu
  • TU Wien, Austria


We thank Nadara et al. [Wojciech Nadara et al., 2019] for publishing their code, some small parts of which we reused in our implementations.

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Alexander Dobler, Manuel Sorge, and Anaïs Villedieu. Turbocharging Heuristics for Weak Coloring Numbers. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Bounded expansion and nowhere-dense classes of graphs capture the theoretical tractability for several important algorithmic problems. These classes of graphs can be characterized by the so-called weak coloring numbers of graphs, which generalize the well-known graph invariant degeneracy (also called k-core number). Being NP-hard, weak-coloring numbers were previously computed on real-world graphs mainly via incremental heuristics. We study whether it is feasible to augment such heuristics with exponential-time subprocedures that kick in when a desired upper bound on the weak coloring number is breached. We provide hardness and tractability results on the corresponding computational subproblems. We implemented several of the resulting algorithms and show them to be competitive with previous approaches on a previously studied set of benchmark instances containing 86 graphs with up to 183831 edges. We obtain improved weak coloring numbers for over half of the instances.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Structural sparsity
  • parameterized algorithms
  • parameterized complexity
  • fixed-parameter tractability


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  1. Faisal N. Abu-Khzam, Shaowei Cai, Judith Egan, Peter Shaw, and Kai Wang. Turbo-charging dominating set with an FPT subroutine: Further improvements and experimental analysis. In T. V. Gopal, Gerhard Jäger, and Silvia Steila, editors, Proceedings of the 14th Annual Conference on Theory and Applications of Models of Computation (TAMC 2017), volume 10185 of Lecture Notes in Computer Science, pages 59-70, 2017. URL: https://doi.org/10.1007/978-3-319-55911-7_5.
  2. 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 2018), pages 143-151. ACM, 2018. URL: https://doi.org/10.1145/3210377.3210383.
  3. Alex Pothen. The complexity of optimal elimination trees, 1988. Google Scholar
  4. Hans L. Bodlaender, Jitender S. Deogun, Klaus. Jansen, Ton. Kloks, Dieter. Kratsch, Haiko. Müller, and Zsolt. Tuza. Rankings of graphs. SIAM Journal on Discrete Mathematics, 11(1):168-181, 1998. URL: https://doi.org/10.1137/S0895480195282550.
  5. Michael Breen-McKay, Brian Lavallee, and Blair D. Sullivan. Hardness of the generalized coloring numbers. CoRR, abs/2112.10562, 2021. URL: http://arxiv.org/abs/2112.10562.
  6. Marco Bressan. Faster algorithms for counting subgraphs in sparse graphs. Algorithmica, 83(8):2578-2605, 2021. URL: https://doi.org/10.1007/s00453-021-00811-0.
  7. Marcin Briański, Piotr Micek, Michał Pilipczuk, and Michał T. Seweryn. Erdös-hajnal properties for powers of sparse graphs. SIAM Journal on Discrete Mathematics, 35(1):447-464, 2021. URL: https://doi.org/10.1137/20M1342756.
  8. Leizhen Cai, Siu Man Chan, and Siu On Chan. Random separation: A new method for solving fixed-cardinality optimization problems. In Proceedings of the Second International Workshop on Parameterized and Exact Computation (IWPEC 2006), pages 239-250. Springer, 2006. URL: https://doi.org/10.1007/11847250_22.
  9. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, 3rd Edition. MIT Press, 2009. URL: http://mitpress.mit.edu/books/introduction-algorithms.
  10. Erik D. Demaine, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, Somnath Sikdar, and Blair D. Sullivan. Structural sparsity of complex networks: Bounded expansion in random models and real-world graphs. Journal of Computer and System Sciences, 105:199-241, 2019. URL: https://doi.org/10.1016/j.jcss.2019.05.004.
  11. Alexander Dobler. Turbocharging heuristics for weak coloring numbers. Master’s thesis, TU Wien, 2021. URL: https://doi.org/10.34726/hss.2021.91580.
  12. Alexander Dobler. Turbocharging heuristics for weak coloring numbers: Source code, November 2021. URL: https://doi.org/10.5281/zenodo.5732923.
  13. Alexander Dobler, Manuel Sorge, and Anaïs Villedieu. Turbocharging heuristics for weak coloring numbers. CoRR, abs/2203.03358, 2022. URL: http://arxiv.org/abs/2203.03358.
  14. R. G. Downey, J. Egan, M. R. Fellows, F. A. Rosamond, and P. Shaw. Dynamic dominating set and turbo-charging greedy heuristics. Tsinghua Science and Technology, 19(4):329-337, 2014. URL: https://doi.org/10.1109/TST.2014.6867515.
  15. Jan Dreier. Lacon- and shrub-decompositions: A new characterization of first-order transductions of bounded expansion classes. In Proceedings of the 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2021), pages 1-13. IEEE, 2021. URL: https://doi.org/10.1109/LICS52264.2021.9470680.
  16. Zdeněk Dvořák. Constant-factor approximation of the domination number in sparse graphs. European Journal of Combinatorics, 34(5):833-840, 2013. URL: https://doi.org/10.1016/j.ejc.2012.12.004.
  17. Kord Eickmeyer, Archontia C. Giannopoulou, Stephan Kreutzer, O.-joung Kwon, Michal Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. Neighborhood complexity and kernelization for nowhere dense classes of graphs. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), pages 63:1-63:14. Schloss Dagstuhlendash Leibniz-Zentrum fuer Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.63.
  18. David Eppstein, Maarten Löffler, and Darren Strash. Listing all maximal cliques in large sparse real-world graphs. Journal of Experimental Algorithmics, 18:3.1:3.1-3.1:3.21, 2013. URL: https://doi.org/10.1145/2543629.
  19. Serge Gaspers, Joachim Gudmundsson, Mitchell Jones, Julián Mestre, and Stefan Rümmele. Turbocharging treewidth heuristics. Algorithmica, 81(2):439-475, 2019. URL: https://doi.org/10.1007/s00453-018-0499-1.
  20. Martin Grohe, Stephan Kreutzer, Roman Rabinovich, Sebastian Siebertz, and Konstantinos S. Stavropoulos. Coloring and covering nowhere dense graphs. SIAM Journal on Discrete Mathematics, 32(4):2467-2481, 2018. URL: https://doi.org/10.1137/18M1168753.
  21. 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. URL: https://doi.org/10.1145/3051095.
  22. Anne-Sophie Himmel, Hendrik Molter, Rolf Niedermeier, and Manuel Sorge. Adapting the bronendash kerbosch algorithm for enumerating maximal cliques in temporal graphs. Social Network Analysis and Mining, 7(1):35, 2017. URL: https://doi.org/10.1007/s13278-017-0455-0.
  23. Falk Hüffner, Christian Komusiewicz, and Manuel Sorge. Finding highly connected subgraphs. In Giuseppe F. Italiano, Tiziana Margaria-Steffen, Jaroslav Pokorný, Jean-Jacques Quisquater, and Roger Wattenhofer, editors, SOFSEM 2015: Theory and Practice of Computer Science, pages 254-265. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-46078-8_21.
  24. Gwenaël Joret, Piotr Micek, Patrice Ossona de Mendez, and Veit Wiechert. Nowhere dense graph classes and dimension. Combinatorica, 39(5):1055-1079, 2019. URL: https://doi.org/10.1007/s00493-019-3892-8.
  25. Meir Katchalski, William McCuaig, and Suzanne Seager. Ordered colourings. Discrete Mathematics, 142(1):141-154, 1995. URL: https://doi.org/10.1016/0012-365X(93)E0216-Q.
  26. Hal A. Kierstead and Daqing Yang. Orderings on graphs and game coloring number. Order, 20(3):255-264, 2003. URL: https://doi.org/10.1023/B:ORDE.0000026489.93166.cb.
  27. Christian Komusiewicz and Manuel Sorge. An algorithmic framework for fixed-cardinality optimization in sparse graphs applied to dense subgraph problems. Discrete Applied Mathematics, 193:145-161, 2015. URL: https://doi.org/10.1016/j.dam.2015.04.029.
  28. Łukasz Kowalik, Marcin Mucha, Wojciech Nadara, Marcin Pilipczuk, Manuel Sorge, and Piotr Wygocki. The pace 2020 parameterized algorithms and computational experiments challenge: Treedepth. In Yixin Cao and Marcin Pilipczuk, editors, Proceedings of the 15th International Symposium on Parameterized and Exact Computation (IPEC 2020), volume 180 of Leibniz International Proceedings in Informatics (LIPIcs), pages 37:1-37:18. Schloss Dagstuhlendash Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.IPEC.2020.37.
  29. O-joung Kwon, Michał Pilipczuk, and Sebastian Siebertz. On low rank-width colorings. European Journal of Combinatorics, 83:103002, 2020. URL: https://doi.org/10.1016/j.ejc.2019.103002.
  30. David W. Matula and Leland L. Beck. Smallest-last ordering and clustering and graph coloring algorithms. Journal of the ACM, 30(3):417-427, 1983. URL: https://doi.org/10.1145/2402.322385.
  31. R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon. Network motifs: Simple building blocks of complex networks. Science, 298(5594):824-827, 2002. URL: https://doi.org/10.1126/science.298.5594.824.
  32. Wojciech Nadara, Marcin Pilipczuk, Roman Rabinovich, Felix Reidl, and Sebastian Siebertz. Empirical evaluation of approximation algorithms for generalized graph coloring and uniform quasi-wideness. ACM Journal of Experimental Algorithmics, 24(1):2.6:1-2.6:34, 2019. URL: https://doi.org/10.1145/3368630.
  33. Jaroslav Nešetřil and Patrice Ossona de Mendez. Sparsity: Graphs, Structures, and Algorithms. Springer, 2012. Google Scholar
  34. Jaroslav Nešetřil, Patrice Ossona de Mendez, Michał Pilipczuk, and Xuding Zhu. Clustering powers of sparse graphs. The Electronic Journal of Combinatorics, pages P4.17-P4.17, 2020. URL: https://doi.org/10.37236/9417.
  35. Michael P. O'Brien and Blair D. Sullivan. An experimental evaluation of a bounded expansion algorithmic pipeline. arXiv:1712.06690 [cs], 2017. URL: http://arxiv.org/abs/1712.06690.
  36. Michał Pilipczuk, Sebastian Siebertz, and Szymon Toruńczyk. Parameterized circuit complexity of model-checking on sparse structures. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2018), pages 789-798. ACM, 2018. URL: https://doi.org/10.1145/3209108.3209136.
  37. Ali Pinar, C. Seshadhri, and Vaidyanathan Vishal. Escape: Efficiently counting all 5-vertex subgraphs. In Proceedings of the 26th International Conference on World Wide Web (WWW 2017), pages 1431-1440. International World Wide Web Conferences Steering Committee, 2017. URL: https://doi.org/10.1145/3038912.3052597.
  38. N. Przulj, D. G. Corneil, and I. Jurisica. Modeling interactome: scale-free or geometric? Bioinformatics, 20(18):3508-3515, 2004. URL: https://doi.org/10.1093/bioinformatics/bth436.
  39. Felix Reidl and Blair D. Sullivan. A color-avoiding approach to subgraph counting in bounded expansion classes. arXiv:2001.05236 [cs], 2020. URL: http://arxiv.org/abs/2001.05236.
  40. Felix Reidl, Fernando Sánchez Villaamil, and Konstantinos Stavropoulos. Characterising bounded expansion by neighbourhood complexity. European Journal of Combinatorics, 75:152-168, 2019. URL: https://doi.org/10.1016/j.ejc.2018.08.001.
  41. Sepp Hartung and Rolf Niedermeier. Incremental list coloring of graphs, parameterized by conservation. Theoretical Computer Science, 494:86-98, 2013. URL: https://doi.org/10.1016/j.tcs.2012.12.049.
  42. Sebastian Siebertz. Nowhere dense graph classes and algorithmic applications. a tutorial at highlights of logic, games and automata 2019. arXiv:1909.06752 [cs], 2019. URL: http://arxiv.org/abs/1909.06752.
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