This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and (∆ + 1)-vertex coloring), the randomized complexity is at most polylogarithmic in the size n of the network, while the best deterministic complexity is typically 2 O( √ log n) . Understanding and potentially narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms.We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in poly log n rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and (∆ + 1)-coloring) in poly log n rounds in the LOCAL model.Perhaps most surprisingly, we show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values.In addition, our formal framework also allows us to develop polylogarithmic-time randomized distributed algorithms in a simpler way. As a result, we provide a polylog-time distributed approximation scheme for arbitrary distributed covering and packing integer linear programs.The question of whether a given distributed problem can be solved locally has been at the center of the theory of distributed graph algorithms since the 1980s, especially starting with the seminal work of Awerbuch, Goldberg, Luby, and Plotkin [AGLP89], Linial [Lin92], and Naor and Stockmeyer [NS95]. The locality of distributed computations is captured by the LOCAL model [Lin92, Pel00], defined as follows: a network is modeled as an undirected graph G = (V, E), the nodes V are the network devices, and the edges E are bidirectional communication links. Time is divided into synchronous communication rounds. In each round, each node can perform some arbitrary internal computation, send a message of possibly arbitrary size to each of its neighbors, and receive the messages sent to it by its neighbors. A typical objective in this setting is to solve some given graph problem on the network G by a distributed algorithm. For example, classic problems include computing a vertex or...

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We present a complete classification of the deterministic distributed time complexity for a family of graph problems: binary labeling problems in trees in the usual LOCAL model of distributed computing. These are locally checkable problems that can be encoded with an alphabet of size two in the edge labeling formalism. Examples of binary labeling problems include sinkless orientation, sinkless and sourceless orientation, 2-vertex coloring, and perfect matching. We show that the complexity of any such problem is in one of the following classes: 𝑂 (1), Θ(log 𝑛), Θ(𝑛), or unsolvable. Furthermore, given the description of any binary labeling problem, we can easily determine in which of the four classes it is and what is an asymptotically optimal algorithm for solving it.

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The algorithmic small-world phenomenon, empirically established by Milgram's letter forwarding experiments from the 60s [59], was theoretically explained by Kleinberg in 2000 [46]. However, from today's perspective his model has several severe shortcomings that limit the applicability to real-world networks. In order to give a more convincing explanation of the algorithmic small-world phenomenon, we study decentralized greedy routing in a more flexible random graph model (geometric inhomogeneous random graphs) which overcomes all previous shortcomings. Apart from exhibiting good properties in theory, it has also been extensively experimentally validated that this model reasonably captures real-world networks.In this model, the greedy routing protocol is purely distributed as each vertex only needs to know information about its direct neighbors. We prove that it succeeds with constant probability, and in case of success almost surely finds an almost shortest path of length Θ(log log n), where our bound is tight including the leading constant. Moreover, we study natural local patching methods which augment greedy routing by backtracking and which do not require any global knowledge. We show that such methods can ensure success probability 1 in an asymptotically tight number of steps.These results also address the question of Krioukov et al. [51] whether there are efficient local routing protocols for the internet graph. There were promising experimental studies, but the question remained unsolved theoretically. Our results give for the first time a rigorous and analytical affirmative answer. *

We develop deterministic approximation algorithms for the minimum dominating set problem in the CONGEST model with an almost optimal approximation guarantee. For ε > 1/ poly log ∆ we obtain two algorithms with approximation factor (1 + ε)(1 + ln(∆ + 1)) and with runtimes 2 O( √ log n log log n) and O(∆ poly log ∆ + poly log ∆ log * n), respectively. Further we show how dominating set approximations can be deterministically transformed into a connected dominating set in the CONGEST model while only increasing the approximation guarantee by a constant factor. This results in a deterministic O(log ∆)-approximation algorithm for the minimum connected dominating set with time complexity 2 O( √ log n log log n) .

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