Simpler and More General Distributed Coloring Based on Simple List Defective Coloring Algorithms

TL;DR

This paper introduces simplified, efficient list defective coloring algorithms that generalize existing methods, enabling faster distributed graph coloring and extending to graphs with bounded neighborhood independence.

Contribution

It presents a new, simpler algorithm for list defective coloring that improves computational efficiency and extends to graphs with bounded neighborhood independence.

Findings

Achieves a more efficient list defective coloring algorithm.

Provides an alternative near-optimal coloring algorithm in the CONGEST model.

Extends coloring techniques to graphs with bounded neighborhood independence.

Abstract

In this paper, we give list coloring variants of simple iterative defective coloring algorithms. Formally, in a list defective coloring instance, each node $v$ of a graph is given a list $L_v$ of colors and a list of allowed defects $d_v(x)$ for the colors. Each node $v$ needs to be colored with a color $x\in L_v$ such that at most $d_v(x)$ neighbors of $v$ also pick the same color $x$. For a defect parameter $d$, it is known that by making two sweeps in opposite order over the nodes of an edge-oriented graph with maximum outdegree $\beta$, one can compute a coloring with $O(\beta^2/d^2)$ colors such that every node has at most $d$ outneighbors of the same color. We generalize this and show that if all nodes have lists of size $p^2$ and $\forall v:\sum_{x\in L_v}(d_v(x)+1)>p\cdot\beta$, we can make two sweeps of the nodes such that at the end, each node $v$ has chosen a color $x\in L_v$ for which at most $d_v(x)$ outneighbors of $v$ are colored with color $x$. Our algorithm is simpler and computationally significantly more efficient than existing algorithms for similar list defective coloring problems. We show that the above result can in particular be used to obtain an alternative $\tilde{O}(\sqrt{\Delta})+O(\log^* n)$-round algorithm for the $(\Delta+1)$-coloring problem in the CONGEST model. The neighborhood independence $\theta$ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. It is known that by doing a single sweep over the nodes of a graph of neighborhood independence $\theta$, one can compute a $d$-defective coloring with $O(\theta\cdot \Delta/d)$ colors. We extend this approach to the list defective coloring setting and use it to obtain an efficient recursive coloring algorithm for graphs of neighborhood independence $\theta$. In particular, if $\theta=O(1)$, we get an $(\log\Delta)^{O(\log\log\Delta)}+O(\log^* n)$-round algorithm.