Optimal short cycle decomposition in almost linear time

Short cycle decomposition is an edge partitioning of an unweighted graph into edge-disjoint short cycles, plus a small number of extra edges not in any cycle. This notion was introduced by Chu et al. [FOCS'18] as a fundamental tool for graph sparsification and sketching. Clearly, it is most desirable to have a fast algorithm for partitioning the edges into as short as possible cycles, while omitting few edges. The most naïve procedure for such decomposition runs in time O(m · n) and partitions the edges into O(log n)-length edge-disjoint cycles plus at most 2n edges. Chu et al. improved the running time considerably to m¹⁺o⁽¹⁾, while increasing both the length of the cycles and the number of omitted edges by a factor of n^o(1). Even more recently, Liu-Sachdeva-Yu [SODA'19] showed that for every constant δ ∈ (0, 1] there is an O(m · n^δ)-time algorithm that provides, w.h.p., cycles of length O(log n)^1/δ and O(n) extra edges. In this paper, we significantly improve upon these bounds. We first show an m¹⁺o⁽¹⁾-time deterministic algorithm for computing nearly optimal cycle decomposition, i.e., with cycle length O(log² n) and an extra subset of O(n log n) edges not in any cycle. This algorithm is based on a reduction to low-congestion cycle covers, introduced by the authors in [SODA'19]. We also provide a simple deterministic algorithm that computes edge-disjoint cycles of length 2¹/ with n¹⁺ · 2^1/ extra edges, for every ∈ (0, 1]. Combining this with Liu-Sachdeva-Yu [SODA'19] gives a linear time randomized algorithm for computing cycles of length poly(log n) and O(n) extra edges, for every n-vertex graphs with n^1+1/δ edges for some constant δ. These decomposition algorithms lead to improvements in all the algorithmic applications of Chu et al. as well as to new distributed constructions.