Distributed Binary Labeling Problems in High-Degree Graphs

Abstract

Balliu et al. (DISC 2020) classified the hardness of solving binary labeling problems with distributed graph algorithms; in these problems the task is to select a subset of edges in a $2$-colored tree in which white nodes of degree $d$ and black nodes of degree $δ$ have constraints on the number of selected incident edges. They showed that the deterministic round complexity of any such problem is $O_{d,δ}(1)$, $Θ_{d,δ}(\log n)$, or $Θ_{d,δ}(n)$, or the problem is unsolvable. However, their classification only addresses complexity as a function of $n$; here $O_{d,δ}$ hides constants that may depend on parameters $d$ and $δ$. In this work we study the complexity of binary labeling problems as a function of all three parameters: $n$, $d$, and $δ$. To this end, we introduce the family of structurally simple problems, which includes, among others, all binary labeling problems in which cardinality constraints can be represented with a context-free grammar. We classify possible complexities of structurally simple problems. As our main result, we show that if the complexity of a problem falls in the broad class of $Θ_{d,δ}(\log n)$, then the complexity for each $d$ and $δ$ is always either $Θ(\log_d n)$, $Θ(\log_δn)$, or $Θ(\log n)$. To prove our upper bounds, we introduce a new, more aggressive version of the rake-and-compress technique that benefits from high-degree nodes.