Graph Sparsification, Spectral Sketches, and Faster Resistance Computation, via Short Cycle Decompositions

Abstract

We develop a framework for graph sparsification and sketching, based on a new tool, short cycle decomposition -a decomposition of an unweighted graph into an edge-disjoint collection of short cycles, plus a small number of extra edges. A simple observation gives that every graph G on n vertices with m edges can be decomposed in O(mn) time into cycles of length at most 2 log n, and at most 2n extra edges. We give an m 1+o(1) time algorithm for constructing a short cycle decomposition, with cycles of length n o(1) , and n 1+o(1) extra edges. Both the existential and algorithmic variants of this decomposition enable us to make progress on several open problems in randomized graph algorithms.1. We present an algorithm that runs in time m 1+o(1) ε −1.5 and returns (1±ε)-approximations to effective resistances of all edges, improving over the previous best of O (min{mε −2 , n 2 ε −1 }). This routine in turn gives an algorithm to approximate the determinant of a graph Laplacian up to a factor of (1 ± ε) in m 1+o(1) + n 15 /8+o(1) ε − 7 /4 time.2. We show existence and efficient algorithms for constructing graphical spectral sketchesa distribution over sparse graphs H such that for a fixed vector x , we havewhere L is the graph Laplacian and L + is its pseudoinverse. This implies the existence of resistance-sparsifiers with about nε −1 edges that preserve the effective resistances between every pair of vertices up to (1 ± ε). By combining short cycle decompositions with known tools in graph sparsification, weshow the existence of nearly-linear sized degree-preserving spectral sparsifiers, as well as significantly sparser approximations of directed graphs. The latter is critical to recent breakthroughs on faster algorithms for solving linear systems in directed Laplacians.The running time and output qualities of our spectral sketch and degree-preserving (directed) sparsification algorithms are limited by the efficiency of our routines for constructing short cycle decompositions. Improved algorithms for short cycle decompositions will lead to improvements for each of these algorithms.2. What if we only want to preserve the effective resistance 2 between all pairs of vertices? Dinitz, Krauthgamer, and Wagner [DKW15] define such a graph H as a resistance sparsifier of G, and show their existence for regular expanders with degree Ω(n). They conjecture that every graph admits an ε-resistance sparsifier with O (nε −1 ) edges.3. An ε-spectral sparsifier preserves weighted vertex degrees up to (1±ε). Do there exist spectral sparsifiers that exactly preserve weighted degrees? Dinitz et al. [DKW15] also explicitly pose a related question -does every dense regular expander contain a sparse regular expander? 4. What about sparsification for directed graphs? The above sparsification notions, and algorithms are difficult to generalize to directed graphs. Cohen et al. [CKP + 16] developed a notion of sparsification for Eulerian directed graphs (directed graphs with all vertices having in-degree equal to out-degree), and gave the first a...

Flow Metrics on Graphs

“…We remark that G ′ does not have to be a subgraph of G, and moreover it can have edge weights. Chu et al [CGPSSW18] show that every graph G with n vertices admits an ε-resistance sparsifier with O(n/ε) edges. We conjecture that this is tight (up to the polylog factors), and prove a weaker lower bound.…”

“…We discuss this result in subsection 3.2. Note that the case of p = 2 is in fact the case of resistance sparsifiers, for which [CGPSSW18] show a better upper bound of O (nε −1 ) edges. We remark that it is an open question to give lower bounds for this problem for p = 2.…”

“…Open Question 3.3. Chu et al [CGPSSW18] show that every graph with n vertices admits an ε-resistance sparsifiers with O (n/ε) edges. Is this tight?…”

“…However in our case, in order to preserve the d ∞ metric, it suffices to preserve only the minimum-st-cuts. In addition, note that the case p = 2 is the special case where d 2 2 is in fact the resistance distance, for which [CGPSSW18] showed the existence of a resistance sparsifier with O (nε −1 ) edges.…”

Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration

“…Although this guarantee suffices for many applications (e.g. [24,9]), some other applications [30,8], including the triangle enumeration algorithm of [7], crucially needs the fact that each part in the decomposition induces an expander. Nanongkai and Saranurak [29] and, independently, Wulff-Nilsen [45] gave a fast algorithm without weakening the guarantee as the one in [41,42].…”