Density Independent Algorithms for Sparsifying k-Step Random Walks

We give faster algorithms for producing sparse approximations of the transition matrices of k-step random walks on undirected and weighted graphs. These transition matrices also form graphs, and arise as intermediate objects in a variety of graph algorithms. Our improvements are based on a better understanding of processes that sample such walks, as well as tighter bounds on key weights underlying these sampling processes. On a graph with n vertices and m edges, our algorithm produces a graph with about nlog(n) edges that approximates the k-step random walk graph in about m + k^2 nlog^4(n) time. In order to obtain this runtime bound, we also revisit "density independent" algorithms for sparsifying graphs whose runtime overhead is expressed only in terms of the number of vertices

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

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 few 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. These decompositions enable us to make progress on several open questions: * We give an algorithm to find $(1\pm\epsilon)$-approximations to effective resistances of all edges in time $m^{1+o(1)}\epsilon^{-1.5}$, improving over the previous best of $\tilde{O}(\min\{m\epsilon^{-2},n^2 \epsilon^{-1}\})$. This gives an algorithm to approximate the determinant of a Laplacian up to $(1\pm\epsilon)$ in $m^{1 + o(1)} + n^{15/8+o(1)}\epsilon^{-7/4}$ time. * We show existence and efficient algorithms for constructing graphical spectral sketches -- a distribution over sparse graphs H such that for a fixed vector $x$, we have w.h.p. $x'L_Hx=(1\pm\epsilon)x'L_Gx$ and $x'L_H^+x=(1\pm\epsilon)x'L_G^+x$. This implies the existence of resistance-sparsifiers with about $n\epsilon^{-1}$ edges that preserve the effective resistances between every pair of vertices up to $(1\pm\epsilon).$ * By combining short cycle decompositions with known tools in graph sparsification, we show 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. Improved algorithms for constructing short cycle decompositions will lead to improvements for each of the above results.Comment: 80 page

Dual algorithms for the densest subgraph problem

Dense subgraph discovery is an important primitive for many real-world graph mining applications. The dissertation tackles the densest subgraph problem via its dual linear programming formulation. Particularly, our contributions in this thesis are the following: (i) We give a faster width-dependent algorithm to solve mixed packing and covering LPs, a class of problems that is fundamental to combinatorial optimization in computer science and operations research (the dual of the densest subgraph problem is an instance of this class of linear programs) . Our work utilizes the framework of area convexity introduced by Sherman [STOC `17] to obtain accelerated rates of convergence. (ii) We devise an iterative algorithm for the densest subgraph problem which naturally generalizes Charikar's greedy algorithm. Our algorithm draws insights from the iterative approaches from convex optimization, and also exploits the dual interpretation of the densest subgraph problem. We have empirical evidence that our algorithm is much more robust against the structural heterogeneities in real-world datasets, and converges to the optimal subgraph density even when the simple greedy algorithm fails. (iii) Lastly, we design the first fully-dynamic algorithm which maintains a $(1-\epsilon)$ approximate densest subgraph in worst-case $\text{poly}(\log n, \epsilon^{-1})$ time per update. Our result improves upon the previous best approximation factor of $(1/4 - \epsilon)$ for fully dynamic densest subgraph.Ph.D