Core Concepts
This work presents deterministic algorithms for efficiently computing vertex colorings in sparse graphs parameterized by their arboricity, a measure of sparsity, in the Adaptive Massively Parallel Computation (AMPC) model.
Abstract
The paper studies the vertex coloring problem in graphs with low arboricity, a measure of sparsity. The key contributions are:
An efficient deterministic Local Computation Algorithm (LCA) that computes a "weak" version of a β-partition, which is a partition of the vertices into layers such that each node has at most β neighbors in higher layers. This LCA uses a sublinear number of queries per node.
Leveraging the LCA, the authors design deterministic AMPC algorithms that compute a complete β-partition with different trade-offs between the number of layers and the runtime.
Using the computed β-partitions, the authors present several AMPC coloring algorithms that provide different guarantees on the number of colors used and the runtime. These include:
An O(α^2+ε)-coloring in O(1/ε) rounds
An O(α^2)-coloring in O(log α) rounds
A ((2+ε)α + 1)-coloring in O(α log α) rounds
The key technical challenge is that the color of a node may depend on almost all other nodes in the graph, and these dependencies cannot be stored on a single machine in the low-space AMPC model. The authors overcome this by carefully exploring the graph structure using their novel LCA.