Short Cycles via Low-Diameter Decompositions

We present improved algorithms for short cycle decomposition of a graph. Short cycle decompositions were introduced in the recent work of Chu et al, and were used to make progress on several questions in graph sparsification. For all constants $\delta \in (0,1]$, we give an $O(mn^\delta)$ time algorithm that, given a graph $G,$ partitions its edges into cycles of length $O(\log n)^\frac{1}{\delta}$, with $O(n)$ extra edges not in any cycle. This gives the first subquadratic, in fact almost linear time, algorithm achieving polylogarithmic cycle lengths. We also give an $m \cdot \exp(O(\sqrt{\log n}))$ time algorithm that partitions the edges of a graph into cycles of length $\exp(O(\sqrt{\log n} \log\log n))$, with $O(n)$ extra edges not in any cycle. This improves on the short cycle decomposition algorithms given in Chu et al in terms of all parameters, and is significantly simpler. As a result, we obtain faster algorithms and improved guarantees for several problems in graph sparsification -- construction of resistance sparsifiers, graphical spectral sketches, degree preserving sparsifiers, and approximating the effective resistances of all edges.