  Given a graph $G=(V,E)$ with arboricity $\alpha$, we study the problem of
decomposing the edges of $G$ into $(1+\epsilon)\alpha$ disjoint forests in the
distributed LOCAL model. Barenboim and Elkin [PODC `08] gave a LOCAL algorithm
that computes a $(2+\epsilon)\alpha$-forest decomposition using $O(\frac{\log
n}{\epsilon})$ rounds. Ghaffari and Su [SODA `17] made further progress by
computing a $(1+\epsilon) \alpha$-forest decomposition in $O(\frac{\log^3
n}{\epsilon^4})$ rounds when $\epsilon \alpha = \Omega(\sqrt{\alpha \log n})$,
i.e. the limit of their algorithm is an $(\alpha+ \Omega(\sqrt{\alpha \log
n}))$-forest decomposition. This algorithm, based on a combinatorial
construction of Alon, McDiarmid \& Reed [Combinatorica `92], in fact provides a
decomposition of the graph into \emph{star-forests}, i.e. each forest is a
collection of stars.
  Our main result in this paper is to reduce the threshold of $\epsilon \alpha$
in $(1+\epsilon)\alpha$-forest decomposition and star-forest decomposition.
This further answers the $10^{\text{th}}$ open question from Barenboim and
Elkin's "Distributed Graph Algorithms" book. Moreover, it gives the first
$(1+\epsilon)\alpha$-orientation algorithms with {\it linear dependencies} on
$\epsilon^{-1}$.
  At a high level, our results for forest-decomposition are based on a
combination of network decomposition, load balancing, and a new structural
result on local augmenting sequences. Our result for star-forest decomposition
uses a more careful probabilistic analysis for the construction of Alon,
McDiarmid, \& Reed; the bounds on star-arboricity here were not previously
known, even non-constructively.
