List Defective Colorings: Distributed Algorithms and Applications

Abstract

The distributed coloring problem is at the core of the area of distributed graph algorithms and it is a problem that has seen tremendous progress over the last few years. Much of the remarkable recent progress on deterministic distributed coloring algorithms is based on two main tools: a) defective colorings in which every node of a given color can have a limited number of neighbors of the same color and b) list coloring, a natural generalization of the standard coloring problem that naturally appears when colorings are computed in different stages and one has to extend a previously computed partial coloring to a full coloring.
In this paper, we introduce list defective colorings, which can be seen as a generalization of these two coloring variants. Essentially, in a list defective coloring instance, each node v is given a list of colors x_{v,1},… ,x_{v,p} together with a list of defects d_{v,1},… ,d_{v,p} such that if v is colored with color x_{v, i}, it is allowed to have at most d_{v, i} neighbors with color x_{v, i}.
We highlight the important role of list defective colorings by showing that faster list defective coloring algorithms would directly lead to faster deterministic (Δ+1)-coloring algorithms in the LOCAL model. Further, we extend a recent distributed list coloring algorithm by Maus and Tonoyan [DISC '20]. Slightly simplified, we show that if for each node v it holds that ∑_{i=1}^p (d_{v,i}+1)² > deg_G²(v)⋅ polylogΔ then this list defective coloring instance can be solved in a communication-efficient way in only O(logΔ) communication rounds. This leads to the first deterministic (Δ+1)-coloring algorithm in the standard CONGEST model with a time complexity of O(√{Δ}⋅ polylog Δ+log^* n), matching the best time complexity in the LOCAL model up to a polylogΔ factor.