Distributed Degree Splitting, Edge Coloring, and Orientations

Abstract

We study a family of closely-related distributed graph problems, which we call degree splitting, where roughly speaking the objective is to partition (or orient) the edges such that each node's degree is split almost uniformly. Our findings lead to answers for a number of problems, a sampling of which includes: -- We present a $poly(\log n)$ round deterministic algorithm for $(2\Delta-1)\cdot (1+o(1))$-edge-coloring, where $\Delta$ denotes the maximum degree. Modulo the $1+o(1)$ factor, this settles one of the long-standing open problems of the area from the 1990's (see e.g. Panconesi and Srinivasan [PODC'92]). Indeed, a weaker requirement of $(2\Delta-1)\cdot poly(\log \Delta)$-edge-coloring in $poly(\log n)$ rounds was asked for in the 4th open question in the Distributed Graph Coloring book by Barenboim and Elkin. -- We show that sinkless orientation---i.e., orienting edges such that each node has at least one outgoing edge---on $\Delta$-regular graphs can be solved in $O(\log_{\Delta} \log n)$ rounds randomized and in $O(\log_{\Delta} n)$ rounds deterministically. These prove the corresponding lower bounds by Brandt et al. [STOC'16] and Chang, Kopelowitz, and Pettie [FOCS'16] to be tight. Moreover, these show that sinkless orientation exhibits an exponential separation between its randomized and deterministic complexities, akin to the results of Chang et al. for $\Delta$-coloring $\Delta$-regular trees. -- We present a randomized $O(\log^4 n)$ round algorithm for orienting $a$-arboricity graphs with maximum out-degree $a(1+\epsilon)$. This can be also turned into a decomposition into $a (1+\epsilon)$ forests when $a=\Omega(\log n)$ and into $a (1+\epsilon)$ pseduo-forests when $a=o(\log n)$. Obtaining an efficient distributed decomposition into less than $2a$ forests was stated as the 10th open problem in the book by Barenboim and Elkin.