The Maximal Independent Set (MIS) problem is one of the basics in the study of locality in distributed graph algorithms. This paper presents an extremely simple randomized algorithm providing a near-optimal local complexity for this problem, which incidentally, when combined with some known techniques, also leads to a near-optimal global complexity.Classical MIS algorithms of Luby [STOC'85] and Alon, Babai and Itai [JALG'86] provide the global complexity guarantee that, with high probability 1 , all nodes terminate after O(log n) rounds. In contrast, our initial focus is on the local complexity, and our main contribution is to provide a very simple algorithm guaranteeing that each particular node v terminates after O(log deg(v) + log 1/ε) rounds, with probability at least 1 − ε. The guarantee holds even if the randomness outside 2-hops neighborhood of v is determined adversarially. This degree-dependency is optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC'04].Interestingly, this local complexity smoothly transitions to a global complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider [FOCS'12; arXiv: 1202], we 2 get a randomized MIS algorithm with a high probability global complexity of O(log ∆) + 2 O( √ log log n) , where ∆ denotes the maximum degree. This improves over the O(log 2 ∆) + 2 O( √ log log n) result of Barenboim et al., and gets close to the Ω(min{log ∆, √ log n}) lower bound of Kuhn et al.Corollaries include improved algorithms for MIS in graphs of upper-bounded arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local Computation Algorithms (LCA) model, and a faster distributed algorithm for the Lovász Local Lemma.1 As standard, we use the phrase with high probability to indicate that an event has probability at least 1 − 1/n. 2 quasi nanos, gigantium humeris insidentes

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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...

The gap between the known randomized and deterministic local distributed algorithms underlies arguably the most fundamental and central open question in distributed graph algorithms. In this paper, we combine the method of conditional expectation with network decompositions to obtain a generic and clean recipe for derandomizing randomized LOCAL algorithms and transforming them into efficient deterministic LOCAL algorithms. This simple recipe leads to significant improvements on a number of problems, in cases resolving known open problems. Two main results are:• An improved deterministic distributed algorithm for hypergraph maximal matching, improving on Fischer, Ghaffari, and Kuhn [FOCS'17], and giving improved algorithms for edge-coloring, maximum matching approximation, and low out-degree edge orientation. The last result gives the first positive resolution in the Open Problem 11.10 in the book of Barenboim and Elkin. • Improved randomized and deterministic distributed algorithms for the Lovász Local Lemma, which gets closer to a conjecture of Chang and Pettie [FOCS'17], and moreover leads to improved distributed algorithms for problems such as defective coloring and k-SAT. * Supported by ERC Grant No. 336495 (ACDC).SLOCAL is similar to the LOCAL model, in that each node can read its r-hop neighborhood in the graph G, for some parameter r. However, in the SLOCAL model, the neighborhoods are read sequentially. Formally, the nodes are processed in an arbitrary (adversarially chosen) order. When node v is processed, v can read its r-hop neighborhood and it computes and locally stores its output y v and potentially additional information. When reading the r-hop neighborhood, v also reads all the information that has been locally stored by the previously processed nodes there. We call the parameter r the locality of an SLOCAL algorithm.The SLOCAL model can be seen as a natural extension of sequential greedy algorithms. In fact, the classic distributed graph problems such as MIS or (∆ + 1)-coloring have simple SLOCAL algorithms with locality 1: in order to determine whether a node v is in the MIS or which color v gets in a (∆ + 1)-coloring, it suffices to know the decisions of all the neighbors that have been processed before.The SLOCAL model is inherently sequential. The main reason it is useful is that there are transformations from SLOCAL algorithms to LOCAL algorithms, which handles symmetry breaking in a "black-box" or generic way. By developing and analyzing SLOCAL algorithms, we are therefore able to treat a number of diverse LOCAL problems in a unified and abstract fashion. We are also able to adapt multiple types of algorithms to take advantage of special structure in the graphs (for example, bounds on its maximum degree). This two-part method of algorithm analysis -constructing SLOCAL algorithms, and transforming them to LOCAL algorithms generically -will be a key technical tool. Our Contributions, Part I: DerandomizationIn the first part of this paper, we present a simple and clean recipe for derandomizing...

We present the first conditional hardness results for massively parallel algorithms for some central graph problems including (approximating) maximum matching, vertex cover, maximal independent set, and coloring. In some cases, these hardness results match or get close to the state of the art algorithms. Our hardness results are conditioned on a widely believed conjecture in massively parallel computation about the complexity of the connectivity problem. We also note that it is known that an unconditional variant of such hardness results might be somewhat out of reach for now, as it would lead to considerably improved circuit complexity lower bounds and would concretely imply that NC 1 ⊊ P. We obtain our conditional hardness result via a general method that lifts unconditional lower bounds from the well-studied LOCAL model of distributed computing to the massively parallel computation setting.

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We introduce a method for "sparsifying" distributed algorithms and exhibit how it leads to improvements that go past known barriers in two algorithmic settings of large-scale graph processing: Massively Parallel Computation (MPC), and Local Computation Algorithms (LCA). • MPC with Strongly Sublinear Memory: Recently, there has been growing interest in obtaining MPC algorithms that are faster than their classic O(log n)-round parallel (PRAM) counterparts for problems such as Maximal Independent Set (MIS), Maximal Matching, 2-Approximation of Minimum Vertex Cover, and (1+ε)-Approximation of Maximum Matching.Currently, all such MPC algorithms require memory ofΩ(n) per machine: Czumaj et al. [STOC'18] were the first to handleΩ(n) memory, running in O((log log n) 2 ) rounds, who improved on the n 1+Ω(1) memory requirement of the O(1)-round algorithm of Lattanzi et al [SPAA'11]. We obtainÕ( √ log ∆)-round MPC algorithms for all these four problems that work even when each machine has strongly sublinear memory, e.g., n α for any constant α ∈ (0, 1). Here, ∆ denotes the maximum degree. These are the first sublogarithmic-time MPC algorithms for (the general case of) these problems that break the linear memory barrier. • LCAs with Query Complexity Below the Parnas-Ron Paradigm: Currently, the best known LCA for MIS has query complexity ∆ O(log ∆) poly(log n), by Ghaffari [SODA'16], which improved over the ∆ O(log 2 ∆) poly(log n) bound of Levi et al. [Algorithmica'17]. As pointed out by Rubinfeld, obtaining a query complexity of poly(∆ log n) remains a central open question. Ghaffari's bound almost reaches a ∆ Ω( log ∆ log log ∆ ) barrier common to all known MIS LCAs, which simulate a distributed algorithm by learning the full local topology,à la Parnas-Ron [TCS'07]. There is a barrier because the distributed complexity of MIS has a lower bound of Ω log ∆ log log ∆ , by results of Kuhn, et al. [JACM '16], which means this methodology cannot go below query complexity ∆ Ω( log ∆ log log ∆ ) . We break this barrier and obtain an LCA for MIS that has a query complexity ∆ O(log log ∆) poly(log n).