Nikhil Srivastava

Graph sparsification by effective resistances

We present a nearly-linear time algorithm that produces high-quality sparsifiers of weighted graphs. Given as input a weighted graph G=(V,E,w) and a parameter ε>0, we produce a weighted subgraph H=(V,~E,~w) of G such that |~E|=O(n log n/ε²) and for all vectors x in R^V. (1-ε) ∑uv ∈ E (x(u)-x(v))²w_uv≤ ∑uv in ~E(x(u)-x(v))²~w_uv ≤ (1+ε)∑uv ∈ E(x(u)-x(v))²w_uv. This improves upon the sparsifiers constructed by Spielman and Teng, which had O(n log^c n) edges for some large constant c, and upon those of Benczur and Karger, which only satisfied (1) for x in {0,1}^V. We conjecture the existence of sparsifiers with O(n) edges, noting that these would generalize the notion of expander graphs, which are constant-degree sparsifiers for the complete graph. A key ingredient in our algorithm is a subroutine of independent interest: a nearly-linear time algorithm that builds a data structure from which we can query the approximate effective resistance between any two vertices in a graph in O(log n) time.

Twice-ramanujan sparsifiers

We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs. In particular, we prove that for every d > 1 and every undirected, weighted graph G = (V,E,w) on n vertices, there exists a weighted graph H=(V,F,~{w}) with at most ⌈d(n-1)⌉ edges such that for every x ∈ R^V, [x^T L_G x ≤ x^T L_H x ≤ ((d+1+2√d)/(d+1-2√d)) • x^T L_G x] where L_G and L_H are the Laplacian matrices of G and H, respectively. Thus, H approximates G spectrally at least as well as a Ramanujan expander with dn/2 edges approximates the complete graph. We give an elementary deterministic polynomial time algorithm for constructing H.

Graph Sparsification by Effective Resistances

We present a nearly linear time algorithm that produces high-quality spectral sparsifiers of weighted graphs. Given as input a weighted graph

Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees

G=(V,E,w)

Twice-Ramanujan Sparsifiers

and a parameter

Spectral sparsification of graphs: theory and algorithms

epsilon>0

Learning and Verifying Graphs Using Queries with a Focus on Edge Counting

, we produce a weighted subgraph

A new approach to computing maximum flows using electrical flows

H=(V,tilde{E},tilde{w})

On the longest path algorithm for reconstructing trees from distance matrices

of

Tight bounds on plurality

G