A Deterministic Distributed $2$-Approximation for Weighted Vertex Cover in $O(\log n\log \mathrm{\Delta }\mathrm{/}{\log }^2\log \mathrm{\Delta })$ Rounds

We present a deterministic distributed $2$-approximation algorithm for the Minimum Weight Vertex Cover problem in the CONGEST model whose round complexity is $O(\log n\log \mathrm{\Delta }\mathrm{/}{\log }^2\log \mathrm{\Delta })$. This improves over the currently best known deterministic 2-approximation implied by [KVY94]. Our solution generalizes the $(2+ϵ)$-approximation algorithm of [BCS17], improving the dependency on ${ϵ}^{-1}$ from linear to logarithmic. In addition, for every $ϵ=(\log \mathrm{\Delta }{)}^{-c}$, where $c\ge 1$ is a constant, our algorithm computes a $(2+ϵ)$-approximation in $O(\log \Delta / \log \log \Delta)$~rounds (which is asymptotically optimal).