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

2025-02-24

AI Summary

This paper addresses the $(Delta+1)$-vertex coloring problem on a graph $G$. We improve upon the classical Palette Sparsification Theorem (PST), whose original proof relies on intricate graph decomposition and requires nontrivial algorithms for coloring after sampling. We introduce the **Asymmetric Palette Sparsification Theorem (APST)**, the first to impose an **average list-length constraint**, guaranteeing colorability with only $O(log n)$ average sample size per vertex and directly supporting standard greedy coloring. This framework substantially simplifies both theoretical analysis and algorithm design, eliminating PST’s technical redundancy and constructive complexity. In semi-streaming, sublinear-time, and Massively Parallel Computation (MPC) models, our approach yields near-optimal $(Delta+1)$-coloring algorithms whose time and space complexities increase by at most $mathrm{polylog}(n)$ factors—achieving both conceptual simplicity and analytical tractability.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Privacy-enhancing technologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Simplifies proofs for $(Δ+1)$ vertex coloring using asymmetric palette sparsification.
• Enables efficient coloring with simpler algorithms in sublinear models.
• Reduces complexity by allowing variable list sizes in vertex coloring.

Innovation

• Asymmetric palette sparsification simplifies vertex coloring.
• Greedy algorithm used for efficient graph coloring.
• Sublinear algorithms achieve near-optimal performance.