Distributed ∆-coloring plays hide-and-seek

We prove several new tight or near-tight distributed lower bounds for classic symmetry breaking problems in graphs. As a basic tool, we first provide a new insightful proof that any deterministic distributed algorithm that computes a Δ-coloring on Δ-regular trees requires Ω(log_Δn) rounds and any randomized such algorithm requires Ω(log_Δlogn) rounds. We prove this by showing that a natural relaxation of the Δ-coloring problem is a fixed point in the round elimination framework.

As a first application, we show that our Δ-coloring lower bound proof directly extends to arbdefective colorings. An arbdefective c-coloring of a graph G=(V,E) is given by a c-coloring of V and an orientation of E, where the arbdefect of a color i is the maximum number of monochromatic outgoing edges of any node of color i. We exactly characterize which variants of the arbdefective coloring problem can be solved in O(f(Δ) + log^*n) rounds, for some function f, and which of them instead require Ω(log_Δn) rounds for deterministic algorithms and Ω(log_Δlogn) rounds for randomized ones.

As a second application, which we see as our main contribution, we use the structure of the fixed point as a building block to prove lower bounds as a function of Δ for problems that, in some sense, are much easier than Δ-coloring, as they can be solved in O(log^* n) deterministic rounds in bounded-degree graphs. More specifically, we prove lower bounds as a function of Δ for a large class of distributed symmetry breaking problems, which can all be solved by a simple sequential greedy algorithm. For example, we obtain novel results for the fundamental problem of computing a (2,β)-ruling set, i.e., for computing an independent set S⊆ V such that every node v∈ V is within distance ≤ β of some node in S. We in particular show that Ω(βΔ^1/β) rounds are needed even if initially an O(Δ)-coloring of the graph is given. With an initial O(Δ)-coloring, this lower bound is tight and without, it still nearly matches the existing O(βΔ^2/(β+1)+log^* n) upper bound. The new (2,β)-ruling set lower bound is an exponential improvement over the best existing lower bound for the problem, which was proven in [FOCS ’20]. As a special case of the lower bound, we also obtain a tight linear-in-Δ lower bound for computing a maximal independent set (MIS) in trees. While such an MIS lower bound was known for general graphs, the best previous MIS lower bounds for trees was Ω(logΔ). Our lower bound even applies to a much more general family of problems that allows for almost arbitrary combinations of natural constraints from coloring problems, orientation problems, and independent set problems, and provides a single unified proof for known and new lower bound results for these types of problems.

All of our lower bounds as a function of Δ also imply substantial lower bounds as a function of n. For instance, we obtain that the maximal independent set problem, on trees, requires Ω(logn / loglogn) rounds for deterministic algorithms, which is tight.