On the Locality of Nash-Williams Forest Decomposition and Star-Forest Decomposition

Given a graph $G=(V,E)$ with arboricity $α$, we study the problem of decomposing the edges of $G$ into $(1+ε)α$ disjoint forests in the distributed LOCAL model. Barenboim and Elkin [PODC `08] gave a LOCAL algorithm that computes a $(2+ε)α$-forest decomposition using $O(\frac{\log n}ε)$ rounds. Ghaffari and Su [SODA `17] made further progress by computing a $(1+ε) α$-forest decomposition in $O(\frac{\log^3 n}{ε^4})$ rounds when $εα= Ω(\sqrt{α\log n})$, i.e. the limit of their algorithm is an $(α+ Ω(\sqrt{α\log n}))$-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 $εα$ in $(1+ε)α$-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 $(1+ε)α$-orientation algorithms with {\it linear dependencies} on $ε^{-1}$. 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.