Improved Distributed Approximation to Maximum Independent Set.

We present improved results for approximating Maximum Independent Set ($\MaxIS$) in the standard LOCAL and CONGEST models of distributed computing. Let $n$ and $\Delta$ 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 $\Delta$-approximation to $\MaxIS$ in $O(\MIS(n,\Delta)\log W)$ rounds, where $\MIS(n,\Delta)$ is the running time for finding a \emph{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 $\MIS$ algorithm that they use as a black-box. Our results: (1) A deterministic $O(\MIS(n,\Delta))$ rounds algorithm for $O(\Delta)$-approximation to $\MaxIS$ in the CONGEST model. (2) A randomized $2^{O(\sqrt{\log \log n})}$ rounds algorithm that finds, with high probability, an $O(\Delta)$-approximation to $\MaxIS$ in the CONGEST model. (3) An $\Omega(\log^*n)$ lower bound for any randomized algorithm that finds an independent set of size $\Omega(n/\Delta)$ that succeeds with probability at least $1-1/\log n$, even for the LOCAL model. This hardness result applies for graphs of maximum degree $\Delta=O(n/\log^*n)$. 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 $\Omega(n/\Delta)$ in graphs with maximum degree $\Delta\leq n/\log n$, with high probability. Due to a lower bound of $\Omega(\sqrt{\log n/\log \log n})$ that was given by Kuhn, Moscibroda and Wattenhofer [JACM, 2016] on the number of rounds for finding a maximal independent set ($\MIS$) in the LOCAL model, even for randomized algorithms, our second result implies that finding an $O(\Delta)$-approximation to $\MaxIS$ is strictly easier than $\MIS$.