Improved Distributed Algorithms for Coloring Interval Graphs with Application to Multicoloring Trees

Abstract

We give a distributed $(1+\epsilon )$-approximation algorithm for the minimum vertex coloring problem on interval graphs, which runs in the $\mathcal {LOCAL}$ model and operates in $\mathrm {O}(\frac{1}{\epsilon } \log ^* n)$ rounds. If nodes are aware of their interval representations, then the algorithm can be adapted to the $\mathcal {CONGEST}$ model using the same number of rounds. Prior to this work, only constant factor approximations using $\mathrm {O}(\log ^* n)$ rounds were known [12]. Linial’s ring coloring lower bound implies that the dependency on $\log ^* n$ cannot be improved. We further prove that the dependency on $\frac{1}{\epsilon }$ is also optimal.

To obtain our $\mathcal {CONGEST}$ model algorithm, we develop a color rotation technique that may be of independent interest. We demonstrate that color rotations can also be applied to obtain a $(1+\epsilon )$-approximate multicoloring of directed trees in $\mathrm {O}( \frac{1}{\epsilon } \log ^* n)$ rounds.

M. M. Halldórsson is supported by grants 152679-05 and 174484-05 from the Icelandic Research Fund. C. Konrad is supported by the Centre for Discrete Mathematics and its Applications (DIMAP) at Warwick University and by EPSRC award EP/N011163/1.