Distributed $Δ$-Coloring Plays Hide-and-Seek

Abstract

We prove several new tight or near-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 ∆-coloring on ∆-regular trees requires Ω(log ∆ n) rounds and any randomized algorithm requires Ω(log ∆ log n) rounds. We prove this result by showing that a natural relaxation of the ∆-coloring problem is a fixed point in the round elimination framework.As a first application, we show that our ∆-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 (∆) + log * n) rounds, for some function f , and which of them instead require Ω(log ∆ n) rounds for deterministic algorithms and Ω(log ∆ log n) rounds for randomized ones.As a second application, which is our main contribution, we use the structure of the fixed point as a building block to prove lower bounds for problems that, in some sense, are much easier than ∆-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 ∆ for a large class of distributed symmetry breaking problems, which can all be solved by a simple sequential greedy algorithm. For example, we obtain a tight linear-in-∆ lower bound for computing a maximal independent set in ∆-regular trees. For the case where an initial O(∆)-coloring is given, we obtain a tight Ω(β∆ 1/β )-round lower bound for computing a (2, β)-ruling set (for β = o(log ∆)). Our lower bounds even apply to a much more general family of problems, such as variants of ruling sets where nodes in the set do not need to satisfy the independence requirement, but they only need to satisfy the requirements of some arbdefective coloring.Our lower bounds as a function of ∆ also imply lower bounds as a function of n. We obtain, for example, that the maximal independent set problem, on trees, requires Ω(log n/ log log n) rounds for deterministic algorithms, which is tight.

The Landscape of Distributed Complexities on Trees and Beyond

“…For the proof of Theorem 3.4, we will make use of an approach that is an extension of the approach known for the setting of LCLs on regular trees without inputs (see, e.g., [5,7,6]). More specifically , we will explicitly define an algorithm A 1 (depending on A) that satisfies the properties stated in Theorem 3.4.…”

Distributed Edge Coloring in Time Polylogarithmic in $Δ$

“…As a result, we therefore also obtain O(∆ + log * n)-round algorithms for MIS and maximal matching. In [3,4,15], it was shown that for MIS and maximal matching, this bound is tight, even on tree networks. More specifically, it was shown that there is no randomized MIS or maximal matching algorithm with round complexity o ∆ + log log n log log log n and there is no deterministic such algorithm with round complexity o ∆ + log n log log n .…”