Short Cycles via Low-Diameter Decompositions

TL;DR

This paper introduces faster algorithms for short cycle decomposition in graphs, achieving near-linear time and improved cycle length bounds, which enhance various graph sparsification techniques.

Contribution

It presents new algorithms for short cycle decomposition with better time complexity and cycle length bounds, simplifying previous methods and improving sparsification applications.

Findings

Achieves near-linear time cycle decomposition with polylogarithmic cycle lengths.

Provides faster algorithms for graph sparsification problems.

Improves guarantees for resistance sparsifiers and spectral sketches.

Abstract

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.