Is a LOCAL algorithm computable?

Abstract

Common definitions of the "standard" LOCAL model tend to be sloppy and even self-contradictory on one point: do the nodes update their state using an arbitrary function or a computable function? So far, this distinction has been safe to neglect, since problems where it matters seem contrived and quite different from e.g. typical local graph problems studied in this context. We show that this question matters even for locally checkable labeling problems (LCLs), perhaps the most widely studied family of problems in the context of the LOCAL model. Furthermore, we show that assumptions about computability are directly connected to another aspect already recognized as highly relevant: whether we have any knowledge of $n$, the size of the graph. Concretely, we show that there is an LCL problem $\Pi$ with the following properties: 1. $\Pi$ can be solved in $O(\log n)$ rounds if the LOCAL model is uncomputable. 2. $\Pi$ can be solved in $O(\log n)$ rounds in the computable model if we know any upper bound on $n$. 3. $\Pi$ requires $\Omega(\sqrt{n})$ rounds in the computable model if we do not know anything about $n$. We also show that the connection between computability and knowledge of $n$ holds in general: for any LCL problem $\Pi$, if you have any bound on $n$, then $\Pi$ has the same round complexity in the computable and uncomputable models.