Simple Sublinear Algorithms for $(\Delta+1)$ Vertex Coloring via Asymmetric Palette Sparsification

TL;DR

This paper introduces an asymmetric palette sparsification theorem that simplifies the process of sublinear graph coloring algorithms, making them easier to implement while maintaining near-optimal performance.

Contribution

The paper presents a weaker version of the palette sparsification theorem allowing different list sizes, enabling simpler and nearly optimal sublinear algorithms for vertex coloring.

Findings

Nearly-optimal sublinear algorithms for $()$ vertex coloring.

Simpler proofs and algorithms compared to previous PST-based methods.

Applicable in semi-streaming, sublinear time, and MPC models.

Abstract

The palette sparsification theorem (PST) of Assadi, Chen, and Khanna (SODA 2019) states that in every graph $G$ with maximum degree $\Delta$, sampling a list of $O(\log{n})$ colors from $\{1,\ldots,\Delta+1\}$ for every vertex independently and uniformly, with high probability, allows for finding a $(\Delta+1)$ vertex coloring of $G$ by coloring each vertex only from its sampled list. PST naturally leads to a host of sublinear algorithms for $(\Delta+1)$ vertex coloring, including in semi-streaming, sublinear time, and MPC models, which are all proven to be nearly optimal, and in the case of the former two are the only known sublinear algorithms for this problem. While being a quite natural and simple-to-state theorem, PST suffers from two drawbacks. Firstly, all its known proofs require technical arguments that rely on sophisticated graph decompositions and probabilistic arguments. Secondly, finding the coloring of the graph from the sampled lists in an efficient manner requires a considerably complicated algorithm. We show that a natural weakening of PST addresses both these drawbacks while still leading to sublinear algorithms of similar quality (up to polylog factors). In particular, we prove an asymmetric palette sparsification theorem (APST) that allows for list sizes of the vertices to have different sizes and only bounds the average size of these lists. The benefit of this weaker requirement is that we can now easily show the graph can be $(\Delta+1)$ colored from the sampled lists using the standard greedy coloring algorithm. This way, we can recover nearly-optimal bounds for $(\Delta+1)$ vertex coloring in all the aforementioned models using algorithms that are much simpler to implement and analyze.