  In this paper, we present a low-diameter decomposition algorithm in the LOCAL
model of distributed computing that succeeds with probability $1 - 1/poly(n)$.
Specifically, we show how to compute an $\left(\epsilon, O\left(\frac{\log
n}{\epsilon}\right)\right)$ low-diameter decomposition in
$O\left(\frac{\log^3(1/\epsilon)\log n}{\epsilon}\right)$ round
  Further developing our techniques, we show new distributed algorithms for
approximating general packing and covering integer linear programs in the LOCAL
model. For packing problems, our algorithm finds an $(1-\epsilon)$-approximate
solution in $O\left(\frac{\log^3 (1/\epsilon) \log n}{\epsilon}\right)$ rounds
with probability $1 - 1/poly(n)$. For covering problems, our algorithm finds an
$(1+\epsilon)$-approximate solution in $O\left(\frac{\left(\log \log n + \log
(1/\epsilon)\right)^3 \log n}{\epsilon}\right)$ rounds with probability $1 -
1/poly(n)$. These results improve upon the previous $O\left(\frac{\log^3
n}{\epsilon}\right)$-round algorithm by Ghaffari, Kuhn, and Maus [STOC 2017]
which is based on network decompositions.
  Our algorithms are near-optimal for many fundamental combinatorial graph
optimization problems in the LOCAL model, such as minimum vertex cover and
minimum dominating set, as their $(1\pm \epsilon)$-approximate solutions
require $\Omega\left(\frac{\log n}{\epsilon}\right)$ rounds to compute.
