The Complexity of Distributed Edge Coloring with Small Palettes

The complexity of distributed edge coloring depends heavily on the palette size as a functionof the maximum degree ∆. In this paper we explore the complexity of edge coloring in theLOCAL model in different palette size regimes. Our results are as follows. • We simplify the round elimination technique of Brandt et al. [9] and prove that (2∆− 2)-edge coloring requires Ω(log∆ log n) time w.h.p. and Ω(log∆ n) time deterministically, evenon trees. The simplified technique is based on two ideas: the notion of an irregular runningtime (in which network components terminate the algorithm at prescribed, but irregulartimes) and some general observations that transform weak lower bounds into stronger ones.• We give a randomized edge coloring algorithm that can use palette sizes as small as ∆ +Õ(√∆), which is a natural barrier for randomized approaches. The running time of thealgorithm is at most O(log ∆·TLLL), where TLLL is the complexity of a permissive versionof the constructive Lovász local lemma.• We develop a new distributed Lovász local lemma algorithm for tree-structured dependencygraphs, which leads to a (1 + )∆-edge coloring algorithm for trees running in O(log log n)time. This algorithm arises from two new results: a deterministic O(log n)-time LLL algo-rithm for tree-structured instances, and a randomized O(log log n)-time graph shatteringmethod for breaking the dependency graph into independent O(log n)-size LLL instances.• A natural approach to computing (∆ + 1)-edge colorings (Vizing’s theorem) is to extendpartial colorings by iteratively re-coloring parts of the graph, e.g., via “augmenting paths.”We prove that this approach may be viable, but in the worst case requires recoloringsubgraphs of diameter Ω(∆ log n). This stands in contrast to distributed algorithms forBrooks’ theorem [34], which exploit the existence of O(log∆ n)-length augmenting paths. ∗Supported by NSF grants CCF-1514383 and CCF-1637546 and ERC Grant No. 336495 (ACDC).arXiv:1708.04290v1[cs.DC]14Aug2017