Consider any locally checkable labeling problem Π in rooted regular trees: there is a finite set of labels Σ, and for each label ∈ Σ we specify what are permitted label combinations of the children for an internal node of label (the leaf nodes are unconstrained). This formalism is expressive enough to capture many classic problems studied in distributed computing, including vertex coloring, edge coloring, and maximal independent set.We show that the distributed computational complexity of any such problem Π falls in one of the following classes: it is (1), Θ(log * ), Θ(log ), or Θ(1) rounds in trees with nodes (and all of these classes are nonempty). We show that the complexity of any given problem is the same in all four standard models of distributed graph algorithms: deterministic LOCAL, randomized LOCAL, deterministic CONGEST, and randomized CONGEST model. In particular, we show that randomness does not help in this setting, and the complexity class Θ(log log ) does not exist (while it does exist in the broader setting of general trees).We also show how to systematically determine the complexity class of any such problem Π, i.e., whether Π takes (1), Θ(log * ), Θ(log ), or Θ(1) rounds. While the algorithm may take exponential time in the size of the description of Π, it is nevertheless practical: we provide a freely available implementation of the classifier algorithm, and it is fast enough to classify many problems of interest.

The locality of a graph problem is the smallest distance T such that each node can choose its own part of the solution based on its radius-T neighborhood. In many settings, a graph problem can be solved efficiently with a distributed or parallel algorithm if and only if it has a small locality. In this work we seek to automate the study of solvability and locality: given the description of a graph problem Π, we would like to determine if Π is solvable and what is the asymptotic locality of Π as a function of the size of the graph. Put otherwise, we seek to automatically synthesize efficient distributed and parallel algorithms for solving Π. We focus on locally checkable graph problems; these are problems in which a solution is globally feasible if it looks feasible in all constant-radius neighborhoods. Prior work on such problems has brought primarily bad news: questions related to locality are undecidable in general, and even if we focus on the case of labeled paths and cycles, determining locality is PSPACE-hard (Balliu et al., PODC 2019). We complement prior negative results with efficient algorithms for the cases of unlabeled paths and cycles and, as an extension, for rooted trees. We study locally checkable graph problems from an automata-theoretic perspective by representing a locally checkable problem Π as a nondeterministic finite automaton M over a unary alphabet. We identify polynomial-time-computable properties of the automaton M that near-completely capture the solvability and locality of Π in cycles and paths, with the exception of one specific case that is co-NP-complete.

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We study the following algorithm synthesis question: given the description of a locally checkable graph problem Π for paths or cycles, determine in which instances Π is solvable, determine what is the distributed round complexity of solving Π in the usual LOCAL model of distributed computing, and construct an asymptotically optimal distributed algorithm for solving Π.To answer such questions, we represent Π as a nondeterministic finite automaton M over a unary alphabet. We classify the states of M into repeatable states, flexible states, mirror-flexible states, loops, and mirror-flexible loops; all of these can be decided in polynomial time. We show that these five classes of states completely answer all questions related to the solvability and distributed computational complexity of Π on cycles.On paths, there is one case in which the question of solvability coincides with the classical universality problem for unary regular languages, and hence determining if a given problem Π is always solvable is co-NP-complete. However, we show that all other questions, including the question of determining the distributed round complexity of Π and finding an asymptotically optimal algorithm for solving Π, can be answered in polynomial time.In prior work, similar questions have been studied in a more general setting, but the results have been mostly negative: for example, determining the distributed complexity of a given locally checkable problem in the case of paths and cycles with input labels is already known to be PSPACEhard (Balliu et al., PODC 2019), and in oriented unlabeled toroidal grids it is undecidable (Brandt et al., PODC 2017). The present work represents the broadest known subclass of distributed algorithm synthesis questions that can be solved efficiently.On the conceptual level, the key contribution is the introduction of a unified automata-theoretic framework for the study of such question, making it possible to also leverage prior work on nondeterministic finite automata. On the technical level, the key new idea is the introduction of the concepts of mirror-flexibility and mirror-flexible loops in automata; we show that these capture exactly those problems that are solvable efficiently with distributed algorithms on undirected paths and cycles.

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