Henrik Lievonen · Timothé Picavet · Jukka Suomela

Distributed binary labeling problems in high-degree graphs

SIROCCO 2024 · 31st International Colloquium on Structural Information and Communication Complexity, Vietri sul Mare, Italy, May 2024

authors’ version arXiv.org


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 $\delta$ have constraints on the number of selected incident edges. They showed that the deterministic round complexity of any such problem is $O_{d,\delta}(1)$, $\Theta_{d,\delta}(\log n)$, or $\Theta_{d,\delta}(n)$, or the problem is unsolvable. However, their classification only addresses complexity as a function of $n$; here $O_{d,\delta}$ hides constants that may depend on parameters $d$ and $\delta$.

In this work we study the complexity of binary labeling problems as a function of all three parameters: $n$, $d$, and $\delta$. 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 $\Theta_{d,\delta}(\log n)$, then the complexity for each $d$ and $\delta$ is always either $\Theta(\log_d n)$, $\Theta(\log_\delta n)$, or $\Theta(\log n)$.

To prove our upper bounds, we introduce a new, more aggressive version of the rake-and-compress technique that benefits from high-degree nodes.

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