Almost Global Problems in the LOCAL Model
International Symposium on Distributed Computing
The landscape of the distributed time complexity is nowadays well-understood for subpolynomial complexities. When we look at deterministic algorithms in the LOCAL model and locally checkable problems (LCLs) in bounded-degree graphs, the following picture emerges: There are lots of problems with time complexities Θ(log * n) or Θ(log n). It is not possible to have a problem with complexity between ω(log * n) and o(log n). In general graphs, we can construct LCL problems with infinitely many
... xities between ω(log n) and n o(1) . In trees, problems with such complexities do not exist. However, the high end of the complexity spectrum was left open by prior work. In general graphs there are problems with complexities of the form Θ(n α ) for any rational 0 < α ≤ 1/2, while for trees only complexities of the form Θ(n 1/k ) are known. No LCL problem with complexity between ω( √ n) and o(n) is known, and neither are there results that would show that such problems do not exist. We show that: In general graphs, we can construct LCL problems with infinitely many complexities between ω( √ n) and o(n). In trees, problems with such complexities do not exist. Put otherwise, we show that any LCL with a complexity o(n) can be solved in time O( √ n) in trees, while the same is not true in general graphs. Recently, in the study of distributed graph algorithms, there has been a lot of interest on structural complexity theory: instead of studying the distributed time complexity of specific graph problems, researchers have started to put more focus on the study of complexity classes in this context. arXiv:1805.04776v2 [cs.DC] 5 Sep 2018 1:2 Almost Global Problems in the LOCAL Model LCL problems. A particularly fruitful research direction has been the study of distributed time complexity classes of so-called LCL problems (locally checkable labellings). We will define LCLs formally in Section 2.2, but the informal idea is that LCLs are graph problems in which feasible solutions can be verified by checking all constant-radius neighbourhoods. Examples of such problems include vertex colouring with k colours, edge colouring with k colours, maximal independent sets, maximal matchings, and sinkless orientations. LCLs play a role similar to the class NP in the centralised complexity theory: these are problems that would be easy to solve with a nondeterministic distributed algorithm -guess a solution and verify it in O(1) rounds -but it is not at all obvious what the distributed time complexity of solving a given LCL problem with deterministic distributed algorithms is. Distributed structural complexity. In the classical (centralised, sequential) complexity theory one of the cornerstones is the time hierarchy theorem  . In essence, it is known that giving more time always makes it possible to solve more problems. Distributed structural complexity is fundamentally different: there are various gap results that establish that there are no LCL problems with complexities in a certain range. For example, it is known that there is no LCL problem whose deterministic time complexity on bounded-degree graphs is between ω(log * n) and o(log n)  . Such gap results have also direct applications: we can speed up algorithms for which the current upper bound falls in one of the gaps. For example, it is known that ∆colouring in bounded-degree graphs can be solved in polylog n time  . Hence 4-colouring in 2-dimensional grids can be also solved in polylog n time. But we also know that in 2-dimensional grids there is a gap in distributed time complexities between ω(log * n) and o( √ n)  , and therefore we know we can solve 4-colouring in O(log * n) time. The ultimate goal here is to identify all such gaps in the landscape of distributed time complexity, for each graph class of interest. State of the art. Some of the most interesting open problems at the moment are related to polynomial complexities in trees. The key results from prior work are: In bounded-degree trees, for each positive integer k there is an LCL problem with time complexity Θ(n 1/k ) . In bounded-degree graphs, for each rational number 0 < α ≤ 1/2 there is an LCL problem with time complexity Θ(n α ) . However, there is no separation between trees and general graphs in the polynomial region. Furthermore, we do not have any LCL problems with time complexities Θ(n α ) for any 1/2 < α < 1. Our contributions. This work resolves both of the above questions. We show that: In bounded-degree graphs, for each rational number 1/2 < α < 1 there is an LCL problem with time complexity Θ(n α ). In bounded-degree trees, there is no LCL problem with time complexity between ω( √ n) and o(n). Hence whenever we have a slightly sublinear algorithm, we can always speed it up to O( √ n) in trees, but this is not always possible in general graphs. Key techniques. We use ideas from the classical centralised complexity theory -e.g. Turing machines and regular languages -to prove results in distributed complexity theory.