# Distributed Δ-Coloring Plays Hide-and-Seek

We prove several new tight distributed lower bounds for classic symmetry breaking graph problems. 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 algorithm requires Ω(log_Δlog n) rounds. We prove this result 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. We exactly characterize which variants of the arbdefective coloring problem are "easy", and which of them instead are "hard". 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 a large class of distributed symmetry breaking problems. For example, we obtain a tight lower bound for the fundamental problem of computing a (2,β)-ruling set. This is an exponential improvement over the best existing lower bound for the problem, which was proven in [FOCS '20]. 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. Our lower bounds as a function of Δ also imply lower bounds as a function of n. We obtain, for example, that maximal independent set, on trees, requires Ω(log n / loglog n) rounds for deterministic algorithms, which is tight.

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