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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

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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