Recurrent Problems in the LOCAL model

Abstract: The paper considers the SUPPORTED model of distributed computing introduced by Schmid and Suomela [HotSDN'13], generalizing the LOCAL and CONGEST models. In this framework, multiple instances of the same problem, differing from each other by the subnetwork to which they apply, recur over time, and need to be solved efficiently online. To do that, one may rely on an initial preprocessing phase for computing some useful information. This preprocessing phase makes it possible, in some cases, to overcome locality-based time lower bounds. A first contribution of the current paper is expanding the spectrum of problem types to which the SUPPORTED model applies. In addition to subnetwork-defined recurrent problems, we introduce also recurrent problems of two additional types: (i) instances defined by partial client sets, and (ii) instances defined by partially fixed outputs. Our second contribution is illustrating the versatility of the SUPPORTED framework by examining recurrent variants of three classical graph problems. The first problem is Minimum Client Dominating Set (CDS), a recurrent version of the classical dominating set problem with each recurrent instance requiring us to dominate a partial client set. We provide a constant time approximation scheme for CDS on trees and planar graphs. The second problem is Color Completion (CC), a recurrent version of the coloring problem in which each recurrent instance comes with a partially fixed coloring (of some of the vertices) that must be completed. We study the minimum number of new colors and the minimum total number of colors necessary for completing this task. The third problem we study is a recurrent version of Locally Checkable Labellings (LCL) on paths of length $n$. We show that such problems have complexities that are either $\Theta(1)$ or $\Theta(n)$, extending the results of Foerster et al. [INFOCOM'19].