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We give a new randomized distributed algorithm for (∆+1)-coloring in the LOCAL model, running√ log log n) rounds in a graph of maximum degree ∆. This implies that the (∆+1)coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of Ω min log n log log n , log ∆ log log ∆ by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to list-coloring where the palette of each node contains ∆ + 1 colors. We extend the set of distributed symmetry-breaking techniques by performing a decomposition of graphs into dense and sparse parts.

The Lovász Local Lemma (LLL), introduced by Erdős and Lovász in 1975, is a powerful tool of the probabilistic method that allows one to prove that a set of n "bad" events do not happen with non-zero probability, provided that the events have limited dependence. However, the LLL itself does not suggest how to find a point avoiding all bad events. Since the work of Beck (1991) there has been a sustained effort to find a constructive proof (i.e. an algorithm) for the LLL or weaker versions of it. In a major breakthrough Moser and Tardos (2010) showed that a point avoiding all bad events can be found efficiently. They also proposed a distributed/parallel version of their algorithm that requires O(log 2 n) rounds of communication in a distributed network.In this paper we provide two new distributed algorithms for the LLL that improve on both the efficiency and simplicity of the Moser-Tardos algorithm. For clarity we express our results in terms of the symmetric LLL though both algorithms deal with the asymmetric version as well. Let p bound the probability of any bad event and d be the maximum degree in the dependency graph of the bad events. When epd 2 < 1 we give a truly simple LLL algorithm running in O(log 1/epd 2 n) rounds. Under the tighter condition ep(d + 1) < 1, we give a slightly slower algorithm running in O(log 2 d • log 1/ep(d+1) n) rounds. Furthermore, we give an algorithm that runs in sublogarithmic rounds under the condition p • f (d) < 1, where f (d) is an exponential function of d. Although the conditions of the LLL are locally verifiable, we prove that any distributed LLL algorithm requires Ω(log * n) rounds.In many graph coloring problems the existence of a valid coloring is established by one or more applications of the LLL.

We study a family of closely-related distributed graph problems, which we call degree splitting, where roughly speaking the objective is to partition (or orient) the edges such that each node's degree is split almost uniformly. Our findings lead to answers for a number of problems, a sampling of which includes:• We present a poly log n round deterministic algorithm for (2∆−1)·(1+o (1) • We show that sinkless orientation-i.e., orienting edges such that each node has at least one outgoing edge-on ∆-regular graphs can be solved in O(log ∆ log n) rounds randomized and in O(log ∆ n) rounds deterministically. These prove the corresponding lower bounds by Brandt et al. [STOC'16] and Chang, Kopelowitz, and Pettie [FOCS'16] to be tight. Moreover, these show that sinkless orientation exhibits an exponential separation between its randomized and deterministic complexities, akin to the results of Chang et al. for ∆-coloring ∆-regular trees.• We present a randomized O(log 4 n) round algorithm for orienting a-arboricity graphs with maximum out-degree a(1 + ε). This can be also turned into a decomposition into a(1 + ε) forests when a = Ω(log n) and into a(1 + ε) pseduo-forests when a = o(log n). Obtaining an efficient distributed decomposition into less than 2a forests was stated as the 10th open problem in the book by Barenboim and Elkin.

The Lovász Local Lemma (LLL), introduced by Erdős and Lovász in 1975, is a powerful tool of the probabilistic method that allows one to prove that a set of n "bad" events do not happen with non-zero probability, provided that the events have limited dependence. However, the LLL itself does not suggest how to find a point avoiding all bad events. Since the work of Beck (1991) there has been a sustained effort to find a constructive proof (i.e. an algorithm) for the LLL or weaker versions of it. In a major breakthrough Moser and Tardos (2010) showed that a point avoiding all bad events can be found efficiently. They also proposed a distributed/parallel version of their algorithm that requires O(log 2 n) rounds of communication in a distributed network.In this paper we provide two new distributed algorithms for the LLL that improve on both the efficiency and simplicity of the Moser-Tardos algorithm. For clarity we express our results in terms of the symmetric LLL though both algorithms deal with the asymmetric version as well. Let p bound the probability of any bad event and d be the maximum degree in the dependency graph of the bad events. When epd 2 < 1 we give a truly simple LLL algorithm running in O(log 1/epd 2 n) rounds. Under the tighter condition ep(d + 1) < 1, we give a slightly slower algorithm running in O(log 2 d · log 1/ep(d+1) n) rounds. Furthermore, we give an algorithm that runs in sublogarithmic rounds under the condition p · f (d) < 1, where f (d) is an exponential function of d. Although the conditions of the LLL are locally verifiable, we prove that any distributed LLL algorithm requires Ω(log * n) rounds. In many graph coloring problems the existence of a valid coloring is established by one or more applications of the LLL. Using our LLL algorithms, we give logarithmic-time *

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Given a weighted bipartite graph, the maximum weight matching (MWM) problem is to find a set of vertex-disjoint edges with maximum weight. We present a new scaling algorithm that runs in O(m √ n log N ) time, when the weights are integers within the range of [0, N ]. The result improves the previous bounds of O(N m √ n) by Gabow and O(m √ n log (nN )) by Gabow and Tarjan over 20 years ago. Our improvement draws ideas from a not widely known result, the primal method by Balinski and Gomory.

Vertex coloring is a central concept in graph theory and an important symmetry-breaking primitive in distributed computing. Whereas degree-Δ graphs may require palettes of Δ +1 colors in the worst case, it is well known that the chromatic number of many natural graph classes can be much smaller. In this paper we give new distributed algorithms to find (Δ/k)-coloring in graphs of girth 4 (triangle-free graphs), girth 5, and trees. The parameter k can be at most (1 4 −o(1)) ln Δ in triangle-free graphs and at most (1 −o(1)) ln Δ in girth-5 graphs and trees, where o(1) is a function of Δ. Specifically, for Δ sufficiently large we can find such a coloring in O (k + log * n) time. Moreover, for any Δ we can compute such colorings in roughly logarithmic time for triangle-free and girth-5 graphs, and in O (log Δ + log Δ log n) time on trees. As a byproduct, our algorithm shows that the chromatic number of triangle-free graphs is at most (4 + o(1)) Δ ln Δ , which improves on Jamall's recent bound of (67 + o(1)) Δ ln Δ. Finally, we show that (Δ + 1)-coloring for triangle-free graphs can be obtained in sublogarithmic time for any Δ.