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].

翻译：我们设计了新的确定性CONGEST近似算法，用于稀疏图中的**最大加权独立集（MWIS）**。作为主要结果，我们获得了新的$\Delta(1+\epsilon)$-近似算法，以及近似比严格依赖于$\alpha$的算法，适用于最大度$\Delta$和树度$\alpha$的图。对于（确定性）$\Delta(1+\epsilon)$-近似，当前最先进的结果来自Faour等人[SODA 2023]近期的一项突破，该研究展示了$O(\log^{2} (\Delta W)\cdot \log (1/\epsilon)+\log ^{*}n)$轮算法，其中$W$是最大节点权重（在常见假设$W=\text{poly}(n)$下，该界限转化为$O(\log^{2} n\cdot\log (1/\epsilon))$）。至于依赖于$\alpha$的近似，通过将上述Faour等人的算法与Kawarabayashi等人[DISC 2020]提出的方法相结合，可以推导出运行时间为$O(\log^{3} n\cdot\log (1/\epsilon))$的确定性CONGEST $(8(1+\epsilon)\cdot\alpha)$-近似算法。