Improved distributed algorithms for coloring interval graphs with application to multicoloring trees.

We give a distributed (1+ϵ)-approximation algorithm for the minimum vertex coloring problem on interval graphs, which runs in the LOCAL model and operates in O(1ϵlog⁎⁡n) rounds. If nodes are aware of their interval representations, then the algorithm can be adapted to the CONGEST model using the same number of rounds. Prior to this work, only constant factor approximations using O(log⁎⁡n) rounds were known [1]. Linial's ring coloring lower bound implies that the dependency on log⁎⁡n cannot be improved. We further prove that the dependency on 1ϵ is also optimal.