Towards Fully Automatic Distributed Lower Bounds

TL;DR

A new and fully automatic method for finding lower bounds of $\Omega(\log_\Delta n)$ and $\Omega(\log_\Delta n)$ rounds for deterministic and randomized algorithms, respectively, via round elimination is developed, and it is shown that this automatic method is indeed useful, by obtaining lower bounds for defective coloring problems.

Abstract

In the past few years, a successful line of research has lead to lower bounds for several fundamental local graph problems in the distributed setting. These results were obtained via a technique called round elimination. On a high level, the round elimination technique can be seen as a recursive application of a function that takes as input a problem $Π$ and outputs a problem $Π'$ that is one round easier than $Π$. Applying this function recursively to concrete problems of interest can be highly nontrivial, which is one of the reasons that has made the technique difficult to approach. The contribution of our paper is threefold. Firstly, we develop a new and fully automatic method for finding lower bounds of $Ω(\log_Δn)$ and $Ω(\log_Δ\log n)$ rounds for deterministic and randomized algorithms, respectively, via round elimination. Secondly, we show that this automatic method is indeed useful, by obtaining lower bounds for defective coloring problems. We show that the problem of coloring the nodes of a graph with $3$ colors and defect at most $(Δ- 3)/2$ requires $Ω(\log_Δn)$ rounds for deterministic algorithms and $Ω(\log_Δ\log n)$ rounds for randomized ones. We note that lower bounds for coloring problems are notoriously challenging to obtain, both in general, and via the round elimination technique. Both the first and (indirectly) the second contribution build on our third contribution -- a new and conceptually simple way to compute the one-round easier problem $Π'$ in the round elimination framework. This new procedure provides a clear and easy recipe for applying round elimination, thereby making a substantial step towards the greater goal of having a fully automatic procedure for obtaining lower bounds in the distributed setting.

Figures (6)

• Figure 1: One possible way to combine $\left\{\mathsf{I}, \mathsf{O}\right\} \left\{\mathsf{I}, \mathsf{O}\right\} \left\{\mathsf{O}\right\}$ with itself. The resulting configuration is $\left\{\mathsf{I}, \mathsf{O}\right\} \left\{\mathsf{I}, \mathsf{O}\right\} \left\{\mathsf{O}\right\}$.
• Figure 2: An example of a valid target diagram, for some problem $\Pi$ with labels $\left\{A,B,X,Y\right\}$. For example, for labels $A$ and $XY$ we have that $\mathrm{Inf}(A,XY) = AXY$ and $\mathrm{Sup}(A,XY) = X$.
• Figure 3: The default diagram for the problem of computing a $1$-defective $2$-coloring in $3$-regular graphs. The symbols $\mspace{2mu}{\color{lightgray}\boxed{\color{black} \cdot}}\mspace{2mu}$ are omitted for clarity, and _ represents the empty set. Edges that can be obtained via transitivity are omitted.
• Figure 4: A collection of combinations that can be used to obtain the configuration $\mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{X}\mathsf{Y}}}\mspace{2mu} ~\mspace{1mu} \mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{X}\mathsf{Y}}}\mspace{2mu} ~\mspace{1mu} \mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{X}\mathsf{Y}}}\mspace{2mu}$. The numbers represent how to match the labels of the two configurations to obtain the parent one. Labels in position $1$ are those where we apply the $\mathrm{Sup}(\cdot,\cdot)$ operator. For example, combining $\mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{A}\mathsf{X}\mathsf{Y}}}\mspace{2mu} ~\mspace{1mu} \mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{B}\mathsf{X}\mathsf{Y}}}\mspace{2mu} ~\mspace{1mu} \mspace{2mu}{\color{lightgray}\boxed{\color{black}}}\mspace{2mu} ~ (2 1 3)$ and $\mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{A}}}\mspace{2mu} ~\mspace{1mu} \mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{A}}}\mspace{2mu} ~\mspace{1mu} \mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{X}}}\mspace{2mu} ~ (1 2 3)$ means taking configuration $\mathrm{Sup}(\mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{B}\mathsf{X}\mathsf{Y}}}\mspace{2mu},\mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{A}}}\mspace{2mu}) ~\mspace{1mu} \mathrm{Inf}(\mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{A}\mathsf{X}\mathsf{Y}}}\mspace{2mu},\mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{A}}}\mspace{2mu}) ~\mspace{1mu} \mathrm{Inf}(\mspace{2mu}{\color{lightgray}\boxed{\color{black}}}\mspace{2mu},\mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{X}}}\mspace{2mu}) = \mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{X}}}\mspace{2mu} ~\mspace{1mu} \mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{A}\mathsf{X}\mathsf{Y}}}\mspace{2mu} ~\mspace{1mu} \mspace{2mu}{\color{lightgray}\boxed{\color{black} \mathsf{X}}}\mspace{2mu}$.
• Figure 5: Diagram $D$ for the case of defective $2$-coloring. Edges that can be obtained via transitivity are omitted.

Theorems & Definitions (38)

• Theorem 3.1
• Corollary 3.2
• Theorem 4.1
• Lemma 4.2: Combination is sound
• proof
• Lemma 4.3: Transitivity
• proof
• Lemma 4.4: Well-foundedness
• proof
• Lemma 4.5: Configuration splitting