There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O(∆ + log * n) communication rounds; here n is the number of nodes and ∆ is the maximum degree. The lower bound by Linial (1987Linial ( , 1992 shows that the dependency on n is optimal: these problems cannot be solved in o(log * n) rounds even if ∆ = 2.However, the dependency on ∆ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds.We prove that the upper bounds are tight. We show that maximal matchings and maximal independent sets cannot be found in o(∆ + log log n/ log log log n) rounds with any randomized algorithm in the LOCAL model of distributed computing.As a corollary, it follows that there is no deterministic algorithm for maximal matchings or maximal independent sets that runs in o(∆ + log n/ log log n) rounds; this is an improvement over prior lower bounds also as a function of n.

Consider a computer network that consists of a path with n nodes. The nodes are labeled with inputs from a constant-sized set, and the task is to find output labels from a constant-sized set subject to some local constraints-more formally, we have an LCL (locally checkable labeling) problem. How many communication rounds are needed (in the standard LOCAL model of computing) to solve this problem?It is well known that the answer is always either O(1) rounds, or Θ(log * n) rounds, or Θ(n) rounds. In this work we show that this question is decidable (albeit PSPACEhard): we present an algorithm that, given any LCL problem defined on a path, outputs the distributed computational complexity of this problem and the corresponding asymptotically optimal algorithm.

In the past few years, a successful line of research has lead to lower bounds for several fundamental local graph problems in the distributed setting. These results were obtained via a technique called round elimination. On a high level, the round elimination technique can be seen as a recursive application of a function that takes as input a problem Π and outputs a problem Π ′ that is one round easier than Π. Applying this function recursively to concrete problems of interest can be highly nontrivial, which is one of the reasons that has made the technique difficult to approach. The contribution of our paper is threefold.Firstly, we develop a new and fully automatic method for finding so-called fixed point relaxations under round elimination. The detection of a non-0-round solvable fixed point relaxation of a problem Π immediately implies lower bounds of Ω(log ∆ n) and Ω(log ∆ log n) rounds for deterministic and randomized algorithms for Π, respectively.Secondly, we show that this automatic method is indeed useful, by obtaining lower bounds for defective coloring problems. More precisely, as an application of our procedure, we show that the problem of coloring the nodes of a graph with 3 colors and defect at most (∆ − 3)/2 requires Ω(log ∆ n) rounds for deterministic algorithms and Ω(log ∆ log n) rounds for randomized ones. Additionally, we provide a simplified proof for an existing defective coloring lower bound. We note that lower bounds for coloring problems are notoriously challenging to obtain, both in general, and via the round elimination technique.Both the first and (indirectly) the second contribution build on our third contribution-a new and conceptually simple way to compute the one-round easier problem Π ′ in the round elimination framework. This new procedure provides a clear and easy recipe for applying round elimination, thereby making a substantial step towards the greater goal of having a fully automatic procedure for obtaining lower bounds in the distributed setting.

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 complexities 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 [12]. 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 whos...

A number of recent papers -e.g.