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:
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.
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