Adaptive Massively Parallel Coloring in Sparse Graphs

Rustam Latypov, Yannic Maus, Shreyas Pai, Jara Uitto

Abstract: Classic symmetry-breaking problems on graphs have gained a lot of attention in models of modern parallel computation. The Adaptive Massively Parallel Computation (AMPC) is a model that captures the central challenges in data center computations. Chang et al. [PODC'2019] gave an extremely fast, constant time, algorithm for the (Δ+1)-coloring problem, where Δ is the maximum degree of an input graph of n nodes. The algorithm works in the most restrictive low-space setting, where each machine has nδ local space for a constant 0 < δ < 1.The standard approaches for (Δ + 1)-coloring are ignorant about the graph topology in the following sense: They exploit the property that any partial coloring can be extended to a feasible (Δ + 1)-coloring of the whole graph. For most graphs, the chromatic number is much smaller than Δ + 1 and we would like to find colorings with fewer colors. However, as soon as we have fewer than Δ + 1 colors, it might not be possible to complete partial colorings.In this work, we study the vertex-coloring problem in sparse graphs parameterized by their arboricity α, a standard measure for sparsity. We give deterministic algorithms that in constant, or almost constant, time give poly α and O(α)-colorings, where α can be arbitrarily smaller than Δ. A strong and standard approach to compute arboricity-dependent colorings is through the Nash-Williams forest decomposition, which gives rise to an (acyclic) orientation of the edges such that each node has a small out-degree.Our main technical contribution is giving efficient deterministic algorithms to compute these orientations and showing how to leverage them to find colorings in low-space AMPC. A key technical challenge is that the color of a node may depend on almost all of the other nodes in the graph and these dependencies cannot be stored on a single machine. Nevertheless, our novel and careful exploration technique yields the orientation, and the arboricity-dependent coloring, with a sublinear number of adaptive queries per node.