  $ \renewcommand{\tilde}{\widetilde} $We present an $\tilde{O}(\log^2 n)$
round deterministic distributed algorithm for the maximal independent set
problem. By known reductions, this round complexity extends also to maximal
matching, $\Delta+1$ vertex coloring, and $2\Delta-1$ edge coloring. These four
problems are among the most central problems in distributed graph algorithms
and have been studied extensively for the past four decades. This improved
round complexity comes closer to the $\tilde{\Omega}(\log n)$ lower bound of
maximal independent set and maximal matching [Balliu et al. FOCS '19]. The
previous best known deterministic complexity for all of these problems was
$\Theta(\log^3 n)$. Via the shattering technique, the improvement permeates
also to the corresponding randomized complexities, e.g., the new randomized
complexity of $\Delta+1$ vertex coloring is now $\tilde{O}(\log^2\log n)$
rounds.
  Our approach is a novel combination of the previously known two methods for
developing deterministic algorithms for these problems, namely global
derandomization via network decomposition (see e.g., [Rozhon, Ghaffari STOC'20;
Ghaffari, Grunau, Rozhon SODA'21; Ghaffari et al. SODA'23]) and local rounding
of fractional solutions (see e.g., [Fischer DISC'17; Harris FOCS'19; Fischer,
Ghaffari, Kuhn FOCS'17; Ghaffari, Kuhn FOCS'21; Faour et al. SODA'23]). We
consider a relaxation of the classic network decomposition concept, where
instead of requiring the clusters in the same block to be non-adjacent, we
allow each node to have a small number of neighboring clusters. We also show a
deterministic algorithm that computes this relaxed decomposition faster than
standard decompositions. We then use this relaxed decomposition to
significantly improve the integrality of certain fractional solutions, before
handing them to the local rounding procedure that now has to do fewer rounding
steps.
