Document Open Access Logo

Distributed (Δ+1)-Coloring in Graphs of Bounded Neighborhood Independence

Authors Marc Fuchs , Fabian Kuhn

PDF

File

Thumbnail PDF
PDF
LIPIcs.OPODIS.2025.23.pdf
  • Filesize: 0.9 MB
  • 21 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Mathematics of computing → Graph coloring
Keywords
  • distributed computing
  • distributed graph algorithms
  • graph coloring
  • list coloring
  • defective coloring

Metrics

Abstract

The distributed coloring problem is arguably one of the key problems studied in the area of distributed graph algorithms. The most standard variant of the problem asks for a proper vertex coloring of a graph with Δ+1 colors, where Δ is the maximum degree of the graph. Despite an immense amount of work on distributed coloring problems in the distributed setting, determining the deterministic complexity of (Δ+1)-coloring in the standard message passing model remains one of the most important open questions of the area. In the LOCAL model, it is known that (Δ+1)-coloring requires Ω(log^* n) rounds even in paths and rings (i.e., when Δ = 2). For general graphs, the problem is known to be solvable in Õ(log^{5/3}n) rounds and in O(√{ΔlogΔ} + log^* n) rounds when expressing the complexity as a function of Δ and with an optimal dependency on n. In the present paper, we aim to improve our understanding of the deterministic complexity of (Δ+1)-coloring as a function of Δ in a special family of graphs for which significantly faster algorithms are already known.
The neighborhood independence θ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. Notable examples of graphs of bounded neighborhood independence are line graphs of graphs and bounded-rank hypergraphs. It is known that the (2Δ-1)-edge coloring problem and therefore the (Δ+1)-coloring problem in line graphs of graphs can be solved in O(log^{12}Δ+log^* n) rounds. In general, in graphs of neighborhood independence θ = O(1), it is known that (Δ+1)-coloring can be solved in 2^{O(√{logΔ})}+O(log^* n) rounds. In the present paper, we significantly improve the latter result, and we show that in graphs of neighborhood independence θ, a (Δ+1)-coloring can be computed in (θ⋅logΔ)^{O(log logΔ / log log logΔ)}+O(log^* n) rounds and thus in quasipolylogarithmic time in Δ as long as θ is at most polylogarithmic in Δ. Our algorithm can be seen as a generalization of an existing similar, but slightly weaker result for (2Δ-1)-edge coloring. We also show that the approach that leads to this polylogarithmic in Δ algorithm for (2Δ-1)-edge coloring already fails for edge colorings of hypergraphs of rank at least 3. At the core of the fast edge coloring algorithm is an algorithm to divide the edges of a graph into two parts so that up to a multiplicative error of 1+o(1), the maximum degree of the line graph induced by each part is at most half the maximum degree of the original line graph. We show that computing such a bipartition of the edges of the line graph of a hypergraph of rank at least 3 requires time logarithmic in n.

Cite As Get BibTex

Marc Fuchs and Fabian Kuhn. Distributed (Δ+1)-Coloring in Graphs of Bounded Neighborhood Independence. In 29th International Conference on Principles of Distributed Systems (OPODIS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 361, pp. 23:1-23:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.OPODIS.2025.23

Author Details

Marc Fuchs
  • University of Freiburg, Germany
Fabian Kuhn
  • University of Freiburg, Germany

Funding

  • Fuchs, Marc: This work was supported by the German Research Foundation (DFG) under the project number 491819048.

References

  1. Baruch Awerbuch, Andrew V. Goldberg, Michael Luby, and Serge A. Plotkin. Network decomposition and locality in distributed computation. In Proc. 30th Symp. on Foundations of Computer Science (FOCS), pages 364-369, 1989. URL: https://doi.org/10.1109/SFCS.1989.63504.
  2. Alkida Balliu, Sebastian Brandt, Juho Hirvonen, Dennis Olivetti, Mikaël Rabie, and Jukka Suomela. Lower bounds for maximal matchings and maximal independent sets. In Proc. 60th IEEE Symp. on Foundations of Computer Science (FOCS), pages 481-497, 2019. URL: https://doi.org/10.1109/FOCS.2019.00037.
  3. Alkida Balliu, Sebastian Brandt, Fabian Kuhn, and Dennis Olivetti. Distributed Δ-coloring plays hide-and-seek. In Proc. 54th ACM Symp. on Theory of Computing (STOC), pages 464-477, 2022. URL: https://doi.org/10.1145/3519935.3520027.
  4. Alkida Balliu, Sebastian Brandt, Fabian Kuhn, and Dennis Olivetti. Distributed edge coloring in time polylogarithmic in Δ. In Proc. 41st ACM Symp. on Principles of Distributed Computing (PODC), pages 15-25, 2022. URL: https://doi.org/10.1145/3519270.3538440.
  5. Alkida Balliu, Mohsen Ghaffari, Fabian Kuhn, and Dennis Olivetti. Node and edge averaged complexities of local graph problems. In Proc. 41st ACM Symp. on Principles of Distributed Computing (PODC), pages 4-14, 2022. URL: https://doi.org/10.1145/3519270.3538419.
  6. Alkida Balliu, Juho Hirvonen, Christoph Lenzen, Dennis Olivetti, and Jukka Suomela. Locality of not-so-weak coloring. In Proc. 26th Int. Coll. on Structural Information and Communication Complexity (SIROCCO), pages 37-51, 2019. URL: https://doi.org/10.1007/978-3-030-24922-9_3.
  7. Alkida Balliu, Fabian Kuhn, and Dennis Olivetti. Distributed edge coloring in time quasi-polylogarithmic in delta. In Proc. 39th ACM Symp. on Principles of Distributed Computing (PODC), pages 289-298, 2020. URL: https://doi.org/10.1145/3382734.3405710.
  8. P. Bamberger, F. Kuhn, and Y. Maus. Efficient deterministic distributed coloring with small bandwidth. In Proc. 39th ACM Symp. on Principles of Distributed Computing (PODC), pages 243-252, 2020. Google Scholar
  9. Leonid Barenboim. Deterministic (Δ+1)-coloring in sublinear (in Δ) time in static, dynamic, and faulty networks. Journal of ACM, 63(5):1-22, 2016. URL: https://doi.org/10.1145/2979675.
  10. Leonid Barenboim and Michael Elkin. Distributed (delta+1)-coloring in linear (in delta) time. In Proc. 41st Annual ACM Symp. on Theory of Computing (STOC), pages 111-120, 2009. URL: https://doi.org/10.1145/1536414.1536432.
  11. Leonid Barenboim and Michael Elkin. Sublogarithmic distributed MIS algorithm for sparse graphs using Nash-Williams decomposition. Distributed Comput., 22:363-379, 2010. URL: https://doi.org/10.1007/s00446-009-0088-2.
  12. Leonid Barenboim and Michael Elkin. Deterministic distributed vertex coloring in polylogarithmic time. Journal of ACM, 58:23:1-23:25, 2011. URL: https://doi.org/10.1145/2027216.2027221.
  13. Leonid Barenboim and Michael Elkin. Distributed deterministic edge coloring using bounded neighborhood independence. In Proc. 30th ACM Symp. on Principles of Distributed Computing (PODC), pages 129-138, 2011. URL: https://doi.org/10.1145/1993806.1993825.
  14. Leonid Barenboim and Michael Elkin. Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers, 2013. URL: https://doi.org/10.2200/S00520ED1V01Y201307DCT011.
  15. Leonid Barenboim, Michael Elkin, and Uri Goldenberg. Locally-iterative distributed (Δ+1)-coloring below Szegedy-Vishwanathan barrier, and applications to self-stabilization and to restricted-bandwidth models. In Proc. 37th ACM Symp. on Principles of Distributed Computing (PODC), pages 437-446, 2018. URL: https://dl.acm.org/citation.cfm?id=3212769.
  16. Leonid Barenboim, Michael Elkin, and Fabian Kuhn. Distributed (Δ+1)-Coloring in Linear (in Δ) Time. SIAM Journal on Computing, 43(1):72-95, 2014. URL: https://doi.org/10.1137/12088848X.
  17. Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider. The Locality of Distributed Symmetry Breaking. Journal of the ACM, 63(3):1-45, 2016. URL: https://doi.org/10.1145/2903137.
  18. Yi-Jun Chang, Qizheng He, Wenzheng Li, Seth Pettie, and Jara Uitto. The complexity of distributed edge coloring with small palettes. In Proc. 29th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 2633-2652, 2018. URL: https://doi.org/10.1137/1.9781611975031.168.
  19. Yi-Jun Chang, Tsvi Kopelowitz, and Seth Pettie. An exponential separation between randomized and deterministic complexity in the LOCAL model. SIAM Journal on Computing, 48(1):122-143, 2019. URL: https://doi.org/10.1137/17M1117537.
  20. Yi-Jun Chang, Wenzheng Li, and Seth Pettie. An optimal distributed (Δ+1)-coloring algorithm? In Proc. 50th ACM Symp. on Theory of Computing, (STOC), pages 445-456, 2018. URL: https://doi.org/10.1145/3188745.3188964.
  21. S. Faour, M. Ghaffari, C. Grunau, F. Kuhn, and V. Rozhon. Local distributed rounding: Generalized to mis, matching, set cover, and beyond. In Proc. of the 2023 ACM-SIAM Symposium on Discrete Algorithms, (SODA 2023), Florence, Italy, January 22-25, 2023, pages 4409-4447, SIAM, 2023. Google Scholar
  22. Ralph L. Faudree, Evelyne Flandrin, and Zdeněk Ryjáček. Claw-free graphs - a survey. Discrete Mathematics, 164(1-3):87-147, 1997. URL: https://doi.org/10.1016/S0012-365X(96)00045-3.
  23. Pierre Fraigniaud, Marc Heinrich, and Adrian Kosowski. Local conflict coloring. In Proc. 57th IEEE Symp. on Foundations of Computer Science (FOCS), pages 625-634, 2016. URL: https://doi.org/10.1109/FOCS.2016.73.
  24. Marc Fuchs and Fabian Kuhn. List defective colorings: Distributed algorithms and applications. In 37th International Symposium on Distributed Computing (DISC 2023), volume 281 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1-22:23, 2023. URL: https://doi.org/10.4230/LIPIcs.DISC.2023.22.
  25. Marc Fuchs and Fabian Kuhn. List defective colorings: Distributed algorithms and applications. CoRR, abs/2304.09666, 2023. URL: https://doi.org/10.48550/arXiv.2304.09666.
  26. Marc Fuchs and Fabian Kuhn. Distributed (δ+1)-coloring in graphs of bounded neighborhood independence. CoRR, abs/2510.21549, 2025. URL: https://arxiv.org/abs/2510.21549.
  27. Mohsen Ghaffari. An improved distributed algorithm for maximal independent set. In Proc. 27th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 270-277, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch20.
  28. Mohsen Ghaffari and Christoph Grunau. Faster deterministic distributed MIS and approximate matching. In Proc. 55th ACM Symp. on Theory of Computing (STOC), pages 1777-1790, 2023. URL: https://doi.org/10.1145/3564246.3585243.
  29. Mohsen Ghaffari and Christoph Grunau. Near-optimal deterministic network decomposition and ruling set, and improved MIS. In Proc. 65th IEEE Symp. on Foundations of Computer Science (FOCS), pages 2148-2179, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00007.
  30. Mohsen Ghaffari, Christoph Grunau, and Václav Rozhon. Improved deterministic network decomposition. In Proc. 32nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2904-2923, 2021. URL: https://doi.org/10.1137/1.9781611976465.173.
  31. Mohsen Ghaffari and Fabian Kuhn. Deterministic distributed vertex coloring: Simpler, faster, and without network decomposition. In Proc. 62nd IEEE Symp. on Foundations of Computing (FOCS), 2021. Google Scholar
  32. A.V. Goldberg, S.A. Plotkin, and G.E. Shannon. Parallel symmetry-breaking in sparse graphs. SIAM Journal on Discrete Mathematics, 1(4):434-446, 1988. URL: https://doi.org/10.1137/0401044.
  33. Magnús M. Halldórsson, Fabian Kuhn, Yannic Maus, and Tigran Tonoyan. Efficient randomized distributed coloring in CONGEST. In Proc. 53rd ACM Symp. on Theory of Computing (STOC), pages 1180-1193, 2021. URL: https://doi.org/10.1145/3406325.3451089.
  34. David G. Harris, Johannes Schneider, and Hsin-Hao Su. Distributed (Δ +1)-coloring in sublogarithmic rounds. J. ACM, 65(4):19:1-19:21, 2018. URL: https://doi.org/10.1145/3178120.
  35. Öjvind Johansson. Simple distributed Delta+1-coloring of graphs. Inf. Process. Lett., 70(5):229-232, 1999. URL: https://doi.org/10.1016/S0020-0190(99)00064-2.
  36. Ken-ichi Kawarabayashi and Gregory Schwartzman. Adapting local sequential algorithms to the distributed setting. In Proc. 32nd Symp. on Distributed Computing (DISC), pages 35:1-35:17, 2018. URL: https://doi.org/10.4230/LIPICS.DISC.2018.35.
  37. Fabian Kuhn. Local weak coloring algorithms and implications on deterministic symmetry breaking. In Proc. 21st ACM Symp. on Parallelism in Algorithms and Architectures (SPAA), 2009. Google Scholar
  38. Fabian Kuhn. Faster deterministic distributed coloring through recursive list coloring. In Proc. 32st ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 1244-1259, 2020. URL: https://doi.org/10.1137/1.9781611975994.76.
  39. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. What cannot be computed locally! In Proc. 23rd ACM Symp. on Principles of Distributed Computing (PODC), pages 300-309, 2004. URL: https://doi.org/10.1145/1011767.1011811.
  40. Fabian Kuhn and Roger Wattenhofer. On the complexity of distributed graph coloring. In Proc. 25th ACM Symp. on Principles of Distributed Computing (PODC), pages 7-15, 2006. URL: https://doi.org/10.1145/1146381.1146387.
  41. Nathan Linial. Distributive graph algorithms – Global solutions from local data. In Proc. 28th Symp. on Foundations of Computer Science (FOCS), pages 331-335. IEEE, 1987. URL: https://doi.org/10.1109/SFCS.1987.20.
  42. L. Lovász. On decompositions of graphs. Studia Sci. Math. Hungar., 1:237-238, 1966. Google Scholar
  43. Michael Luby. A simple parallel algorithm for the maximal independent set problem. SIAM Journal on Computing, 15(4):1036-1053, 1986. URL: https://doi.org/10.1137/0215074.
  44. Yannic Maus and Tigran Tonoyan. Local conflict coloring revisited: Linial for lists. In Proc. 34th Symp. on Distributed Computing (DISC), volume 179 of LIPIcs, pages 16:1-16:18, 2020. URL: https://doi.org/10.4230/LIPICS.DISC.2020.16.
  45. Moni Naor. A lower bound on probabilistic algorithms for distributive ring coloring. SIAM Journal on Discrete Mathematics, 4(3):409-412, 1991. URL: https://doi.org/10.1137/0404036.
  46. Alessandro Panconesi and Aravind Srinivasan. On the complexity of distributed network decomposition. Journal of Algorithms, 20(2):356-374, 1996. URL: https://doi.org/10.1006/JAGM.1996.0017.
  47. David Peleg. Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics, USA, 2000. URL: https://doi.org/10.1137/1.9780898719772.
  48. Václav Rozhoň and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In Proc. 52nd ACM Symp. on Theory of Computing (STOC), pages 350-363, 2020. Google Scholar
  49. Johannes Schneider and Roger Wattenhofer. A new technique for distributed symmetry breaking. In Proc. 29th ACM Symp. on Principles of Distributed Computing (PODC), pages 257-266, 2010. URL: https://doi.org/10.1145/1835698.1835760.
  50. Mario Szegedy and Sundar Vishwanathan. Locality based graph coloring. In Proc. 25th ACM Symp. on Theory of Computing (STOC), pages 201-207, 1993. URL: https://doi.org/10.1145/167088.167156.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact