Distributed Δ-Coloring Plays Hide-and-Seek
Year: 2026
Type: article
Abstract: 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 Δ n ) rounds and any randomized such algorithm requires Ω (log Δ log n ) 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 Δ n ) rounds for deterministic algorithms and Ω (log Δ log n ) 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 (β Δ 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 (β Δ 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 (log Δ). 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 (log n / log log n ) rounds for deterministic algorithms, which is tight.
Source: Journal of the ACM
DOI: 10.1145/3819817
Authors: Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivetti
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W3203451712