Document

# Palette Sparsification Beyond (Δ+1) Vertex Coloring

## File

LIPIcs.APPROX-RANDOM.2020.6.pdf
• Filesize: 0.66 MB
• 22 pages

## 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.

## Cite As

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

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## 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.
3. Noga Alon and Joel Spencer. The Probabilistic Method. Fourth Edition, Wiley, 2016.
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.
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.
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.
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.
8. Anton Bernshteyn. The Johansson-Molloy theorem for DP-coloring. Random Structures & Algorithms, 54(4):653-664, 2019.
9. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM J. Comput., 39(2):546-563, 2009.
10. Béla Bollobás. Chromatic number, girth and maximal degree. Discrete Mathematics, 24(3):311-314, 1978.
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.
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.
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.
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.
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.
17. Oded Goldreich. Introduction to Property Testing. Cambridge University Press, 2017.
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.
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.
20. Penny E Haxell. A note on vertex list colouring. Combinatorics, Probability and Computing, 10(4):345-347, 2001.
21. Anders Johansson. Asymptotic choice number for triangle free graphs. Technical report, unpublished manuscript, 1996.
22. Anders Johansson. The choice number of sparse graphs. Technical report, unpublished manuscript, 1996.
23. Jeong Han Kim. On Brooks' theorem for sparse graphs. Combinatorics, Probability & Computing, 4:97-132, 1995.
24. Michael Molloy. The list chromatic number of graphs with small clique number. J. Comb. Theory, Ser. B, 134:264-284, 2019.
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.
26. Michael Molloy and Bruce A. Reed. Graph coloring and the probabilistic method. New York I Springer, 2002.
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.
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.
29. Seth Pettie and Hsin-Hao Su. Distributed coloring algorithms for triangle-free graphs. Inf. Comput., 243:263-280, 2015.
30. Bruce A. Reed. ω, Δ, and χ. Journal of Graph Theory, 27(4):177-212, 1998.
31. Bruce A. Reed. The list colouring constants. Journal of Graph Theory, 31(2):149-153, 1999.
32. Bruce A. Reed and Benny Sudakov. Asymptotically the list colouring constants are 1. J. Comb. Theory, Ser. B, 86(1):27-37, 2002.
33. Vojtech Rödl. On a packing and covering problem. Eur. J. Comb., 6(1):69-78, 1985.