Improved Deterministic Distributed Maximum Weight Independent Set Approximation in Sparse Graphs

We design new deterministic CONGEST approximation algorithms for \emph{maximum weight independent set (MWIS)} in \emph{sparse graphs}. As our main results, we obtain new $\Delta(1+\epsilon)$-approximation algorithms as well as algorithms whose approximation ratio depend strictly on $\alpha$, in graphs with maximum degree $\Delta$ and arboricity $\alpha$. For (deterministic) $\Delta(1+\epsilon)$-approximation, the current state-of-the-art is due to a recent breakthrough by Faour et al.\ [SODA 2023] that showed an $O(\log^{2} (\Delta W)\cdot \log (1/\epsilon)+\log ^{*}n)$-round algorithm, where $W$ is the largest node-weight (this bound translates to $O(\log^{2} n\cdot\log (1/\epsilon))$ under the common assumption that $W=\text{poly}(n)$). As for $\alpha$-dependent approximations, a deterministic CONGEST $(8(1+\epsilon)\cdot\alpha)$-approximation algorithm with runtime $O(\log^{3} n\cdot\log (1/\epsilon))$ can be derived by combining the aforementioned algorithm of Faour et al.\ with a method presented by Kawarabayashi et al.\ [DISC 2020].