Distributed deterministic edge coloring using bounded neighborhood independence

Abstract

We study the edge-coloring problem in the message-passing model of distributed computing. This is one of the most fundamental problems in this area. Currently, the best-known deterministic algorithms for (2Δ −1)-edge-coloring requires O(Δ) + log* n time (Panconesi and Rizzi in Distrib Comput 14(2):97–100, 2001), where Δ is the maximum degree of the input graph. Also, recent results of Barenboim and Elkin (2010) for vertex-coloring imply that one can get an O(Δ)-edge-coloring in $${O(\Delta^{\epsilon}\cdot \log n)}$$ time, and an $${O(\Delta^{1 + \epsilon})}$$ -edge-coloring in O(log Δ log n) time, for an arbitrarily small constant $${\epsilon > 0}$$ . In this paper we devise a significantly faster deterministic edge-coloring algorithm. Specifically, our algorithm computes an O(Δ)-edge-coloring in $${O(\Delta^{\epsilon}) + \log* n}$$ time, and an $${O(\Delta^{1 + \epsilon})}$$ -edge-coloring in O(log Δ) + log* n time. This result improves the state-of-the-art running time for deterministic edge-coloring with this number of colors in almost the entire range of maximum degree Δ. Moreover, it improves it exponentially in a wide range of Δ, specifically, for 2 Ω(log*n) ≤ Δ ≤ polylog(n). In addition, for small values of Δ (up to log1 - δ n, for some fixed δ > 0) our deterministic algorithm outperforms all the existing randomized algorithms for this problem. Also, our algorithm is the first O(Δ)-edge-coloring algorithm that has running time o(Δ) + log* n, for the entire range of Δ. All previous (deterministic and randomized) O(Δ)-edge-coloring algorithms require $${\Omega(\min \{\Delta, \sqrt{\log n}\ \})}$$ time. On our way to these results we study the vertex-coloring problem on graphs with bounded neighborhood independence. This is a large family of graphs, which strictly includes line graphs of r-hypergraphs (i.e., hypergraphs in which each hyperedge contains r or less vertices) for r = O(1), and graphs of bounded growth. We devise a very fast deterministic algorithm for vertex-coloring graphs with bounded neighborhood independence. This algorithm directly gives rise to our edge-coloring algorithms, which apply to general graphs. Our main technical contribution is a subroutine that computes an O(Δ/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p 2) + log* n time, for a parameter p, 1 ≤ p ≤ Δ. In all previous efficient distributed routines for m-defective p-coloring the product m· p is super-linear in Δ. In our routine this product is linear in Δ, and this enables us to speed up the algorithm drastically.