Faster Algorithms and Graph Structure via Gaussian Elimination

Graph partitioning has played an important role in theoretical computer science, particularly/nin the design of approximation algorithms and metric embeddings. In some of these applica-/ntions, fundamental tradeoffs in graph partitioning prevented further progress. To overcome/nthese barriers, we consider partitions of certain derived graphs of an undirected graph G ob-/ntained by applying Gaussian elimination to the Laplacian matrix of G to eliminate vertices/nfrom G. We use this technique and others to obtain new results on the following fronts:/nCheeger’s Inequality: Cheeger’s inequality shows that any undirected graph G with min-/nimum/n√ nonzero normalized Laplacian eigenvalue λ G has a cut with conductance at most/nO( λ G ). Qualitatively, Cheeger’s inequality says that if the relaxation time of a graph is/nhigh, there is a cut that certifies this. However, there is a gap in this relationship, as cuts/ncan have conductance as low as Θ(λ G )./nTo better approximate the relaxation time of a graph, we consider a more general object./nSpecifically, instead of bounding the mixing time with cuts, we bound it with cuts in graphs/nobtained via Gaussian elimination from G. Combinatorially, random walks in these graphs/nare equivalent in distribution to random walks in G restricted to a subset of its vertices. As/na result, all Schur complement cuts have conductance at least Ω(λ G ). We show that unlike/nwith cuts, this inequality is tight up to a constant factor. Specifically, there is a derived/ngraph containing a cut with conductance at most O(λ G )./nOblivious Routing: We show that in any graph, the average length of a flow path in an/nelectrical flow between the endpoints of a random edge is O(log 2 n). This is a consequence of/na more general result which shows that the spectral norm of the entrywise absolute value of/nthe transfer impedance matrix of a graph is O(log 2 n). This result implies a simple oblivious/nrouting scheme based on electrical flows in the case of transitive graphs./nRandom Spanning Tree Sampling: We give an m 1+o(1) β o(1) -time algorithm for generat-/ning uniformly random spanning trees in weighted graphs with max-to-min weight ratio β. In2/nthe process, we illustrate how fundamental tradeoffs in graph partitioning can be overcome/nby eliminating vertices from a graph using Schur complements of the associated Laplacian/nmatrix./nOur starting point is the Aldous-Broder algorithm, which samples a random spanning tree/nusing a random walk. As in prior work, we use fast Laplacian linear system solvers to shortcut/nthe random walk from a vertex v to the boundary of a set of vertices assigned to v called a/n“shortcutter.” We depart from prior work by introducing a new way of employing Laplacian/nsolvers to shortcut the walk. To bound the amount of shortcutting work, we show that most/nrandom walk steps occur far away from an unvisited vertex. We apply this observation by/ncharging uses of a shortcutter S to random walk steps in the Schur complement obtained by/neliminating all vertices in S that are not assigned to it./nSpectral Sparsification: We introduce a new approach to spectral sparsification that/napproximates the quadratic form of the pseudoinverse of a graph Laplacian restricted to a/nsubspace. We show that sparsifiers with a near-linear number of edges in the dimension/nof the subspace exist. Our setting generalizes that of Schur complement sparsifiers. Our/napproach produces sparsifiers by sampling a uniformly random spanning tree of the input/ngraph and using that tree to guide an edge elimination procedure that contracts, deletes,/nand reweights edges. In the context of Schur complement sparsifiers, our approach has two/nbenefits over prior work. First, it produces a sparsifier in almost-linear time with no runtime/ndependence on the desired error. We directly exploit this to compute approximate effective/nresistances for a small set of vertex pairs in faster time than prior work [33]. Secondly,/nit yields sparsifiers that are reweighted minors of the input graph. As a result, we give a/nnear-optimal answer to a variant of the Steiner point removal problem./nA key ingredient of our algorithm is a subroutine of independent interest: a near-linear/ntime algorithm that, given a chosen set of vertices, builds a data structure from which we/ncan query a multiplicative approximation to the decrease in the effective resistance between/ntwo vertices after identifying all vertices in the chosen set to a single vertex with inverse/npolynomial additional additive error in near-constant time.

Advisor: Rao, Satish
Srivastava, Nikhil