Fast Parallel Degree+1 List Coloring

Graph coloring problems are arguably among the most fundamental graph problems in parallel and distributed computing with numerous applications. In particular, in recent years the classical ($\Delta+1$)-coloring problem became a benchmark problem to study the impact of local computation for parallel and distributed algorithms. In this work, we study the parallel complexity of a generalization of the ($\Delta+1$)-coloring problem: the problem of (degree+1)-list coloring (${\mathsf{D1LC}}$), where each node has an input palette of acceptable colors, of size one more than its degree, and the objective is to find a proper coloring using these palettes. In a recent work, Halld\'orsson et al. (STOC'22) presented a randomized $O(\log^3\log n)$-rounds distributed algorithm for ${\mathsf{D1LC}}$ in the ${\mathsf{LOCAL}}$ model, matching for the first time the state-of-the art complexity for $(\Delta+1)$-coloring due to Chang et al. (SICOMP'20). In this paper, we obtain a similar connection for $\mathsf{D1LC}$ in the Massively Parallel Computation (${\mathsf{MPC}}$) model with sublinear local space: we present a randomized $O(\log\log\log n)$-round ${\mathsf{MPC}}$ algorithm for ${\mathsf{D1LC}}$, matching the state-of-the art ${\mathsf{MPC}}$ algorithm for the $(\Delta+1)$-coloring problem. We also show that our algorithm can be efficiently derandomized.