Local Conflict Coloring Revisited: Linial for Lists

TL;DR

This paper extends Linial's color reduction algorithm to list coloring in directed graphs, achieving faster, more efficient algorithms with smaller message sizes, and improves existing local list coloring bounds.

Contribution

It introduces a list coloring framework that generalizes Linial's algorithm, providing a two-round solution with smaller lists and message sizes, and enhances the state-of-the-art for $(deg+1)$-list coloring.

Findings

Two-round list coloring algorithm with smaller lists.

Reduced message size from large to roughly $eta$.

Improved runtime for $(deg+1)$-list coloring to $O( oot{eta} ext{log}eta)+ ext{log}^* n$.

Abstract

Linial's famous color reduction algorithm reduces a given $m$-coloring of a graph with maximum degree $\Delta$ to a $O(\Delta^2\log m)$-coloring, in a single round in the LOCAL model. We show a similar result when nodes are restricted to choose their color from a list of allowed colors: given an $m$-coloring in a directed graph of maximum outdegree $\beta$, if every node has a list of size $\Omega(\beta^2 (\log \beta+\log\log m + \log \log |\mathcal{C}|))$ from a color space $\mathcal{C}$ then they can select a color in two rounds in the LOCAL model. Moreover, the communication of a node essentially consists of sending its list to the neighbors. This is obtained as part of a framework that also contains Linial's color reduction (with an alternative proof) as a special case. Our result also leads to a defective list coloring algorithm. As a corollary, we improve the state-of-the-art truly local $(deg+1)$-list coloring algorithm from Barenboim et al. [PODC'18] by slightly reducing the runtime to $O(\sqrt{\Delta\log\Delta})+\log^* n$ and significantly reducing the message size (from huge to roughly $\Delta$). Our techniques are inspired by the local conflict coloring framework of Fraigniaud et al. [FOCS'16].