Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs

TL;DR

This paper presents near-optimal distributed algorithms for approximating minimum vertex coloring and maximum independent set in chordal graphs, utilizing tree decompositions to efficiently partition and process the graph layers.

Contribution

It introduces deterministic distributed algorithms with tight bounds for coloring and independent set approximation in chordal graphs, leveraging novel tree decomposition techniques.

Findings

Coloring algorithm runs in $O(rac{1}{psilon} \, \log n)$ rounds.

Independent set algorithm runs in $O(rac{1}{psilon} \, \log(\frac{1}{psilon}) \, \log^* n)$ rounds.

Lower bounds match the algorithm complexities, showing optimality.

Abstract

We give deterministic distributed $(1+\epsilon)$-approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in $O(\frac{1}{\epsilon} \log n)$ rounds, and our independent set algorithm has a runtime of $O(\frac{1}{\epsilon}\log(\frac{1}{\epsilon})\log^* n)$ rounds. For coloring, existing lower bounds imply that the dependencies on $\frac{1}{\epsilon}$ and $\log n$ are best possible. For independent set, we prove that $O(\frac{1}{\epsilon})$ rounds are necessary. Both our algorithms make use of a tree decomposition of the input chordal graph. They iteratively peel off interval subgraphs, which are identified via the tree decomposition of the input graph, thereby partitioning the vertex set into $O(\log n)$ layers. For coloring, each interval graph is colored independently, which results in various coloring conflicts between the layers. These conflicts are then resolved in a separate phase, using the particular structure of our partitioning. For independent set, only the first $O( \log \frac{1}{\epsilon})$ layers are required as they already contain a large enough independent set. We develop a $(1+\epsilon)$-approximation maximum independent set algorithm for interval graphs, which we then apply to those layers. This work raises the question as to how useful tree decompositions are for distributed computing.