On the locality of nash-williams forest decomposition and star-forest decomposition

Given a graph G = (V, E) with arboricity a, we study the problem of decomposing the edges of G into (1+\varepsilon)a disjoint forests in the distributed LOCAL model. Here G may be a simple graph or multigraph. While there is a polynomial time centralized algorithm for a-forest decomposition (e.g., [H. Imai, J. Oper. Res. Soc. Japan, 26 (1983), pp. 186-211]), it remains an open question how close we can get to this exact decomposition in the LOCAL model. Barenboim and Elkin [L. Barenboim and M. Elkin, Sublogarithmic distributed MIS algorithm for sparse graphs using Nash Williams decomposition, Distrib. Comput., 22 (2010), pp. 363--379] developed a LOCAL algorithm to compute a (2+\varepsilon)a-forest decomposition in O(log n \varepsilon ) rounds. Ghaffari and Su [Proc. 28th ACM-SIAM Symposium on Discrete Algorithms, 2017, pp. 2505--2523] made further progress by computing a (1 + \varepsilon)a-forest decomposition in O(log3 n \varepsilon4) rounds when \varepsilon a = S2(\surda log n); i.e., the limit of their algorithm is an (a + S2 (\surda log n))-forest decomposition. This algorithm, based on a combinatorial construction of Alon, McDiarmid, and Reed [Combinatorica, 12 (1992), pp. 375--380], in fact provides a decomposition of the graph into star-forests, i.e., each forest is a collection of stars. Our main goal is to reduce the threshold of \varepsilon a in (1 +\varepsilon)a-forest decomposition. We obtain a number of results with different parameters; some notable examples are the following: (1) An O( & DBLBOND;\rho log4 n \varepsilon )-round algorithm when \varepsilon a = S2\rho(1) in multigraphs, where \rho > 0 is any arbitrary constant; (2) an O(log4 n log & DBLBOND; \varepsilon )round algorithm when \varepsilon a = S2( log & DBLBOND; log log & DBLBOND; ) in multigraphs; (3) an O(log4\varepsilon n )-round algorithm when \varepsilon a = S2(logn) in multigraphs (this also covers an extension of the forest-decomposition problem to list-edge coloring); (4) an O( log3 n \varepsilon )-round algorithm for star-forest decomposition for \varepsilon a = S2(\surd log \Delta + log a) in simple graphs (when \varepsilon a \geq S2(log \Delta ), this also covers a list-coloring variant). Our techniques also give an algorithm for (1 +\varepsilon)a-outdegree-orientation in O( log3 n \varepsilon ) rounds, which is the first algorithm with linear dependency on \varepsilon 1. At a high level, the first three results come from a combination of network decomposition, load balancing, and a new structural result on local augmenting sequences. The fourth result uses a more careful probabilistic analysis for the construction of Alon, McDiarmid, and Reed; the bounds on star-forest decomposition were not previously known even non constructively.