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Palette Sparsification Beyond (Δ+1) Vertex Coloring

Authors Noga Alon, Sepehr Assadi

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Author Details

Noga Alon
  • Department of Mathematics, Princeton University, NJ, USA
  • Schools of Mathematics and Computer Science, Tel Aviv University, Israel
Sepehr Assadi
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA

Funding

  • Alon, Noga: Supported in part by NSF grant DMS-1855464 and the Simons Foundation.
  • Assadi, Sepehr: Part of this work was done while the author was a postdoctoral researcher at Princeton University and was supported in part by Simons Collaboration on Algorithms and Geometry.

Acknowledgements

Sepehr Assadi would like to thank Suman Bera, Amit Chakrabarti, Prantar Ghosh, Guru Guruganesh, David Harris, Sanjeev Khanna, and Hsin-Hao Su for helpful conversations and Mohsen Ghaffari for communicating the (deg+1) coloring problem and an illuminating discussion that led us to the proof of the palette sparsification theorem for this problem in this paper. We are also thankful to the anonymous reviewers of RANDOM 2020 for helpful suggestions.

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Noga Alon and Sepehr Assadi. Palette Sparsification Beyond (Δ+1) Vertex Coloring. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 6:1-6:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.6

Abstract

A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA'19] states that in every n-vertex graph G with maximum degree Δ, sampling O(log n) colors per each vertex independently from Δ+1 colors almost certainly allows for proper coloring of G from the sampled colors. Besides being a combinatorial statement of its own independent interest, this theorem was shown to have various applications to design of algorithms for (Δ+1) coloring in different models of computation on massive graphs such as streaming or sublinear-time algorithms. 
In this paper, we focus on palette sparsification beyond (Δ+1) coloring, in both regimes when the number of available colors is much larger than (Δ+1), and when it is much smaller. In particular,  
- We prove that for (1+ε) Δ coloring, sampling only O_ε(√{log n}) colors per vertex is sufficient and necessary to obtain a proper coloring from the sampled colors - this shows a separation between (1+ε) Δ and (Δ+1) coloring in the context of palette sparsification. 
- A natural family of graphs with chromatic number much smaller than (Δ+1) are triangle-free graphs which are O(Δ/ln Δ) colorable. We prove a palette sparsification theorem tailored to these graphs: Sampling O(Δ^γ + √{log n}) colors per vertex is sufficient and necessary to obtain a proper O_γ(Δ/ln Δ) coloring of triangle-free graphs. 	
- We also consider the "local version" of graph coloring where every vertex v can only be colored from a list of colors with size proportional to the degree deg(v) of v. We show that sampling O_ε(log n) colors per vertex is sufficient for proper coloring of any graph with high probability whenever each vertex is sampling from a list of (1+ε) ⋅ deg(v) arbitrary colors, or even only deg(v)+1 colors when the lists are the sets {1,…,deg(v)+1}.  
Our new palette sparsification results naturally lead to a host of new and/or improved algorithms for vertex coloring in different models including streaming and sublinear-time algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Mathematics of computing → Graph coloring
  • Mathematics of computing → Graph algorithms
Keywords
  • Graph coloring
  • palette sparsification
  • sublinear algorithms
  • list-coloring

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References

  1. Noga Alon and Sepehr Assadi. Palette sparsification beyond (Δ+1) vertex coloring. CoRR, abs/2006.10456, 2020. URL: http://arxiv.org/abs/2006.10456.
  2. Noga Alon, Michael Krivelevich, and Benny Sudakov. Coloring graphs with sparse neighborhoods. J. Comb. Theory, Ser. B, 77(1):73-82, 1999. Google Scholar
  3. Noga Alon and Joel Spencer. The Probabilistic Method. Fourth Edition, Wiley, 2016. Google Scholar
  4. Amihood Amir, Oren Kapah, Tsvi Kopelowitz, Moni Naor, and Ely Porat. The family holiday gathering problem or fair and periodic scheduling of independent sets. In Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2016, Asilomar State Beach/Pacific Grove, CA, USA, July 11-13, 2016, pages 367-375, 2016. Google Scholar
  5. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Sublinear algorithms for (Δ + 1) vertex coloring. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 767-786, 2019. Google Scholar
  6. Nikhil Bansal, Anupam Gupta, and Guru Guruganesh. On the Lovász theta function for independent sets in sparse graphs. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 193-200, 2015. Google Scholar
  7. Suman K. Bera, Amit Chakrabarti, and Prantar Ghosh. Graph coloring via degeneracy in streaming and other space-conscious models. CoRR, abs/1905.00566. To appear in ICALP 2020, 2019. Google Scholar
  8. Anton Bernshteyn. The Johansson-Molloy theorem for DP-coloring. Random Structures & Algorithms, 54(4):653-664, 2019. Google Scholar
  9. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM J. Comput., 39(2):546-563, 2009. Google Scholar
  10. Béla Bollobás. Chromatic number, girth and maximal degree. Discrete Mathematics, 24(3):311-314, 1978. Google Scholar
  11. Marthe Bonamy, Tom Kelly, Peter Nelson, and Luke Postle. Bounding χ by a fraction of Δ for graphs without large cliques. arXiv preprint, 2018. URL: http://arxiv.org/abs/1803.01051.
  12. Yi-Jun Chang, Manuela Fischer, Mohsen Ghaffari, Jara Uitto, and Yufan Zheng. The complexity of (Δ+1) coloring in congested clique, massively parallel computation, and centralized local computation. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC 2019, Toronto, ON, Canada, July 29 - August 2, 2019, pages 471-480, 2019. Google Scholar
  13. Yi-Jun Chang, Wenzheng Li, and Seth Pettie. An optimal distributed (Δ+1)-coloring algorithm? In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 445-456, 2018. Google Scholar
  14. Ewan Davies, Rémi de Joannis de Verclos, Ross J. Kang, and François Pirot. Colouring triangle-free graphs with local list sizes. CoRR, abs/1812.01534, 2018. Google Scholar
  15. Michael Elkin, Seth Pettie, and Hsin-Hao Su. (2Δ-1)-edge-coloring is much easier than maximal matching in the distributed setting. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 355-370, 2015. Google Scholar
  16. Paul Erdős, Arthur L Rubin, and Herbert Taylor. Choosability in graphs. In Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing, Congressus Numerantium, volume 26, pages 125-157, 1979. Google Scholar
  17. Oded Goldreich. Introduction to Property Testing. Cambridge University Press, 2017. Google Scholar
  18. David G Harris, Johannes Schneider, and Hsin-Hao Su. Distributed (Δ+ 1)-coloring in sublogarithmic rounds. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 465-478. ACM, 2016. Google Scholar
  19. Nicholas J. A. Harvey, Christopher Liaw, and Paul Liu. Greedy and local ratio algorithms in the mapreduce model. In Proceedings of the 30th on Symposium on Parallelism in Algorithms and Architectures, SPAA 2018, July 16-18, 2018, pages 43-52, 2018. Google Scholar
  20. Penny E Haxell. A note on vertex list colouring. Combinatorics, Probability and Computing, 10(4):345-347, 2001. Google Scholar
  21. Anders Johansson. Asymptotic choice number for triangle free graphs. Technical report, unpublished manuscript, 1996. Google Scholar
  22. Anders Johansson. The choice number of sparse graphs. Technical report, unpublished manuscript, 1996. Google Scholar
  23. Jeong Han Kim. On Brooks' theorem for sparse graphs. Combinatorics, Probability & Computing, 4:97-132, 1995. Google Scholar
  24. Michael Molloy. The list chromatic number of graphs with small clique number. J. Comb. Theory, Ser. B, 134:264-284, 2019. Google Scholar
  25. Michael Molloy and Bruce A. Reed. A bound on the strong chromatic index of a graph. J. Comb. Theory, Ser. B, 69(2):103-109, 1997. Google Scholar
  26. Michael Molloy and Bruce A. Reed. Graph coloring and the probabilistic method. New York I Springer, 2002. Google Scholar
  27. Merav Parter. (Δ+1) coloring in the congested clique model. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 160:1-160:14, 2018. Google Scholar
  28. Merav Parter and Hsin-Hao Su. Randomized (Δ+1)-coloring in O(log^*Δ) congested clique rounds. In 32nd International Symposium on Distributed Computing, DISC 2018, New Orleans, LA, USA, October 15-19, 2018, pages 39:1-39:18, 2018. Google Scholar
  29. Seth Pettie and Hsin-Hao Su. Distributed coloring algorithms for triangle-free graphs. Inf. Comput., 243:263-280, 2015. Google Scholar
  30. Bruce A. Reed. ω, Δ, and χ. Journal of Graph Theory, 27(4):177-212, 1998. Google Scholar
  31. Bruce A. Reed. The list colouring constants. Journal of Graph Theory, 31(2):149-153, 1999. Google Scholar
  32. Bruce A. Reed and Benny Sudakov. Asymptotically the list colouring constants are 1. J. Comb. Theory, Ser. B, 86(1):27-37, 2002. Google Scholar
  33. Vojtech Rödl. On a packing and covering problem. Eur. J. Comb., 6(1):69-78, 1985. Google Scholar
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