# Improved Distributed Approximation to Maximum Independent Set

We present improved results for approximating Maximum Independent Set () in the standard LOCAL and CONGEST models of distributed computing. Let n and Δ be the number of nodes and maximum degree in the input graph, respectively. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a Δ-approximation to in O((n,Δ) W) rounds, where (n,Δ) is the running time for finding a maximal independent set, and W is the maximum weight of a node in the network. Whether their algorithm is randomized or deterministic depends on the algorithm that they use as a black-box. Our results: (1) A deterministic O((n,Δ)) rounds algorithm for O(Δ)-approximation to in the CONGEST model. (2) A randomized 2^O(√( n)) rounds algorithm that finds, with high probability, an O(Δ)-approximation to in the CONGEST model. (3) An Ω(^*n) lower bound for any randomized algorithm that finds an independent set of size Ω(n/Δ) that succeeds with probability at least 1-1/ n, even for the LOCAL model. This hardness result applies for graphs of maximum degree Δ=O(n/^*n). One might wonder whether the same hardness result applies for low degree graphs. We rule out this possibility with our next result. (4) An O(1) rounds algorithm that finds an independent set of size Ω(n/Δ) in graphs with maximum degree Δ≤ n/ n, with high probability. Due to a lower bound of Ω(√( n/ n)) that was given by Kuhn, Moscibroda and Wattenhofer [JACM, 2016] on the number of rounds for finding a maximal independent set () in the LOCAL model, even for randomized algorithms, our second result implies that finding an O(Δ)-approximation to is strictly easier than .

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