Distributed Binary Labeling Problems in High-Degree Graphs

Henrik Lievonen, Timothé Picavet, Jukka Suomela

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 \(\delta \) have constraints on the number of selected incident edges. They showed that the deterministic round complexity of any such problem is \(\mathcal {O}_{d,\delta }(1)\), \(\varTheta _{d,\delta }(\log n)\), or \(\varTheta _{d,\delta }(n)\), or the problem is unsolvable. However, their classification only addresses complexity as a function of n; here \(\mathcal {O}_{d,\delta }\) 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 \(\varTheta _{d,\delta }(\log n)\), then the complexity for each d and \(\delta \) is always either \(\varTheta (\log _d n)\), \(\varTheta (\log _\delta n)\), or \(\varTheta (\log n)\). To prove our upper bounds, we introduce a new, more aggressive version of the rake-and-compress technique that benefits from high-degree nodes.