[Paper Review] Palette Sparsification Beyond $(\Delta+1)$ Vertex Coloring

Noga Alon, Sepehr Assadi|arXiv (Cornell University)|Jun 18, 2020

Complexity and Algorithms in Graphs29 references4 citations

TL;DR

This paper extends palette sparsification beyond the classic $(\Delta+1)$ vertex coloring framework, proving that for $(1+\varepsilon)\Delta$ coloring, sampling $O_\varepsilon(\sqrt{\log n})$ colors per vertex suffices for proper coloring with high probability. It further establishes sparsification theorems for triangle-free graphs and local list-coloring settings, enabling improved sublinear and streaming algorithms for graph coloring.

ABSTRACT

A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA'19] states that in every $n$-vertex graph $G$ with maximum degree $\Delta$, sampling $O(\log{n})$ colors per each vertex independently from $\Delta+1$ 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 $(\Delta+1)$ coloring in different models of computation on massive graphs such as streaming or sublinear-time algorithms. In this paper, we further study palette sparsification problems: * We prove that for $(1+\varepsilon) \Delta$ coloring, sampling only $O_{\varepsilon}(\sqrt{\log{n}})$ colors per vertex is sufficient and necessary to obtain a proper coloring from the sampled colors. * A natural family of graphs with chromatic number much smaller than $(\Delta+1)$ are triangle-free graphs which are $O(\frac{\Delta}{\ln{\Delta}})$ colorable. We prove that sampling $O(\Delta^{\gamma} + \sqrt{\log{n}})$ colors per vertex is sufficient and necessary to obtain a proper $O_{\gamma}(\frac{\Delta}{\ln{\Delta}})$ coloring of triangle-free graphs. * We show that sampling $O_{\varepsilon}(\log{n})$ colors per vertex is sufficient for proper coloring of any graph with high probability whenever each vertex is sampling from a list of $(1+\varepsilon) \cdot deg(v)$ arbitrary colors, or even only $deg(v)+1$ colors when the lists are the sets $\{1,\ldots,deg(v)+1\}$. Similar to previous work, 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.

Motivation & Objective

• To extend the palette sparsification framework beyond the $(\Delta+1)$ coloring setting to cases where the number of available colors is significantly more or less than $\Delta+1$.
• To analyze the sparsification threshold for $(1+\varepsilon)\Delta$ coloring, showing it requires only $O_\varepsilon(\sqrt{\log n})$ sampled colors per vertex.
• To develop a tailored palette sparsification theorem for triangle-free graphs, which are $O(\Delta / \ln \Delta)$-colorable, requiring $O(\Delta^\gamma + \sqrt{\log n})$ sampled colors per vertex.
• To establish a local palette sparsification result where each vertex uses a list of colors proportional to its degree, proving $O_\varepsilon(\log n)$ sampled colors suffice for $(1+\varepsilon)\cdot \deg(v)$ lists or $\deg(v)+1$ lists.
• To derive new sublinear and streaming algorithms for graph coloring based on the new sparsification results, improving upon prior work in space and query complexity.

Proposed method

• Prove that sampling $O_\varepsilon(\sqrt{\log n})$ colors per vertex from a $(1+\varepsilon)\Delta$ palette ensures a proper coloring with high probability using probabilistic analysis and concentration bounds.
• Analyze the structure of triangle-free graphs using known chromatic number bounds ($O(\Delta / \ln \Delta)$) and apply sparsification under this constraint, proving sufficiency of $O(\Delta^\gamma + \sqrt{\log n})$ sampled colors.
• Introduce a local palette sparsification model where each vertex $v$ samples from a list of size $\approx \deg(v)$, and prove that $O_\varepsilon(\log n)$ sampled colors per vertex suffice for proper coloring with high probability.
• Use random partitioning of vertices into $k$ sets to reduce maximum degree and neighborhood density, enabling recursive coloring with bounded palette size.
• Leverage Chernoff bounds and union bounds to control the number of edges in neighborhoods and degrees in induced subgraphs after random partitioning.
• Derive algorithmic implications by combining sparsification with known sublinear algorithm frameworks, leading to improved space and query complexity in streaming and query models.

Experimental results

Research questions

• RQ1 What is the minimum number of colors that must be sampled per vertex to ensure $(1+\varepsilon)\Delta$ coloring with high probability?
• RQ2 Can palette sparsification be effectively extended to graphs with chromatic number significantly below $\Delta+1$, such as triangle-free graphs?
• RQ3 Does local list-coloring with lists of size $O(\deg(v))$ admit a palette sparsification theorem with sub-logarithmic sampling per vertex?
• RQ4 Can the new sparsification theorems be leveraged to design improved sublinear algorithms for graph coloring in streaming or query models?
• RQ5 What is the trade-off between the number of sampled colors per vertex and the chromatic number in sparse neighborhood or triangle-free graphs?

Key findings

• For $(1+\varepsilon)\Delta$ coloring, sampling $O_\varepsilon(\sqrt{\log n})$ colors per vertex is both sufficient and necessary to achieve a proper coloring with high probability.
• For triangle-free graphs, which are $O(\Delta / \ln \Delta)$-colorable, sampling $O(\Delta^\gamma + \sqrt{\log n})$ colors per vertex is sufficient and necessary for proper $O(\Delta^\gamma / \ln \Delta)$ coloring.
• In the local list-coloring model, sampling $O_\varepsilon(\log n)$ colors per vertex suffices for proper coloring when each vertex samples from a list of size $(1+\varepsilon)\cdot \deg(v)$ colors.
• For the special case where each vertex has a list of exactly $\deg(v)+1$ colors, such as $\{1, \dots, \deg(v)+1\}$, sampling $O_\varepsilon(\log n)$ colors per vertex still ensures a proper coloring with high probability.
• The new sparsification theorems lead to a randomized single-pass streaming algorithm using $\widetilde{O}(n/\delta)$ space for $O(\Delta / \ln(1/\delta))$ coloring in graphs with $\delta$-sparse neighborhoods.
• A non-adaptive query algorithm is obtained with $\widetilde{O}(n^{3/2}/\delta)$ queries and runtime $\widetilde{O}(n^{3/2} \cdot \text{poly}(1/\delta))$ for the same class of graphs.