Rumor Spreading with No Dependence on Conductance

National Science Foundation (U.S.) (CCF-0843915

A simple, combinatorial algorithm for solving SDD systems in nearly-linear time

Original manuscript January 28, 2013In this paper, we present a simple combinatorial algorithm that solves symmetric diagonally dominant (SDD) linear systems in nearly-linear time. It uses little of the machinery that previously appeared to be necessary for a such an algorithm. It does not require recursive preconditioning, spectral sparsification, or even the Chebyshev Method or Conjugate Gradient. After constructing a "nice" spanning tree of a graph associated with the linear system, the entire algorithm consists of the repeated application of a simple update rule, which it implements using a lightweight data structure. The algorithm is numerically stable and can be implemented without the increased bit-precision required by previous solvers. As such, the algorithm has the fastest known running time under the standard unit-cost RAM model. We hope the simplicity of the algorithm and the insights yielded by its analysis will be useful in both theory and practice.National Science Foundation (U.S.) (Award 0843915)National Science Foundation (U.S.) (Award 1111109)Alfred P. Sloan Foundation (Research Fellowship)National Science Foundation (U.S.). Graduate Research Fellowship Program (Grant 1122374

Fitting a graph to vector data

We introduce a measure of how well a combinatorial graph ts a collection of vectors. The optimal graphs under this measure may be computed by solving convex quadratic programs and have many interesting properties. For vectors in d dimensional space, the graphs always have average degree at most 2(d+1), and for vectors in 2 dimensions they are always planar. We compute these graphs for many standard data sets and show that they can be used to obtain good solutions to classifi cation, regression and clustering problems.National Science Foundation (U.S.) (Grant No. CCF-0634957)National Science Foundation (U.S.) (Grant No.CCF-0843915

Electric routing and concurrent flow cutting

We investigate an oblivious routing scheme, amenable to distributed computation and resilient to graph changes, based on electrical flow. Our main technical contribution is a new rounding method which we use to obtain a bound on the operator norm of the inverse graph Laplacian. We show how this norm reflects both latency and congestion of electric routing. Keywords: Oblivious routing; Spectral graph theory; Laplacian operato

Metric uniformization and spectral bounds for graphs

We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate 'Riemannian' metric to uniformize the geometry of the graph. In many interesting cases, the existence of such a metric is shown by examining the combinatorics of special types of flows. This involves proving new inequalities on the crossing number of graphs. In particular, we use our method to show that for any positive integer k, the k [superscript th] smallest eigenvalue of the Laplacian on an n-vertex, bounded-degree planar graph is O(k/n). This bound is asymptotically tight for every k, as it is easily seen to be achieved for square planar grids. We also extend this spectral result to graphs with bounded genus, and graphs which forbid fixed minors. Previously, such spectral upper bounds were only known for the case k = 2.National Science Foundation (U.S.) (NSF grant CCF-0843915)National Science Foundation (U.S.) (NSF grant CCF-0915251)National Science Foundation (U.S.) (grant CCF-0644037)National Science Foundation (U.S.) (Graduate Research Fellowship)Akamai Technologies, Inc.Alfred P. Sloan Foundation (Research Fellowship)National Science Foundation (U.S.) (NSF grant CCF-0635102)National Science Foundation (U.S.) (grant CCF-0964481)National Science Foundation (U.S.) (CCF-1111270

Local graph partitions for approximation and testing

We introduce a new tool for approximation and testing algorithms called partitioning oracles. We develop methods for constructing them for any class of bounded-degree graphs with an excluded minor, and in general, for any hyperfinite class of bounded-degree graphs. These oracles utilize only local computation to consistently answer queries about a global partition that breaks the graph into small connected components by removing only a small fraction of the edges. We illustrate the power of this technique by using it to extend and simplify a number of previous approximation and testing results for sparse graphs, as well as to provide new results that were unachievable with existing techniques. For instance:1. We give constant-time approximation algorithms for the size of the minimum vertex cover, the minimum dominating set, and the maximum independent set for any class of graphs with an excluded minor.2. We show a simple proof that any minor-closed graph property is testable in constant time in the bounded degree model.3. We prove that it is possible to approximate the distance to almost any hereditary property in any bounded degree hereditary families of graphs. Hereditary properties of interest include bipartiteness, k-colorability, and perfectness.National Science Foundation (U.S.) (0732334)National Science Foundation (U.S.) (0728645)National Science Foundation (U.S.) (CCF-0843915)National Science Foundation (U.S.) (CCF-0832997)Symantec Research Labs Graduate FellowshipW. M. Keck Foundation Center for Extreme Quantum Information TheoryAkamai Technologies, Inc

Higher eigenvalues of graphs

We present a general method for proving upper bounds on the eigenvalues of the graph Laplacian. In particular, we show that for any positive integer k, the kth smallest eigenvalue of the Laplacian on a bounded-degree planar graph is O(k/n). This bound is asymptotically tight for every k, as it is easily seen to be achieved for planar grids. We also extend this spectral result to graphs with bounded genus, graphs which forbid fixed minors, and other natural families. Previously, such spectral upper bounds were only known for k = 2, i.e. for the Fiedler value of these graphs. In addition, our result yields a new, combinatorial proof of the celebrated result of Korevaar in differential geometry.National Science Foundation (U.S.) (CCF-0843915)National Science Foundation (U.S.) (CCF-0644037)National Science Foundation (U.S.) (CCF-0635102)Akamai Technologies, Inc.National Science Foundation (U.S.). Graduate Research Fellowship Progra

A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

We prove that with high probability over the choice of a random graph G from the Erds-Rényi distribution G(n,1/2), the n[superscript o(d)]-time degree d Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least n[superscript 1/2-c(d/log n)1/2] for some constant c > 0. This yields a nearly tight n[superscript 1/2-o(1))] bound on the value of this program for any degree d = o(log n). Moreover we introduce a new framework that we call pseudo-calibration to construct Sum-of-Squares lower bounds. This framework is inspired by taking a computational analogue of Bayesian probability theory. It yields a general recipe for constructing good pseudo-distributions (i.e., dual certificates for the Sum-of-Squares semidefinite program), and sheds further light on the ways in which this hierarchy differs from others

Faster Algorithms for Computing the Stationary Distribution, Simulating Random Walks, and More

In this paper, we provide faster algorithms for computing variousfundamental quantities associated with random walks on a directedgraph, including the stationary distribution, personalized PageRankvectors, hitting times, and escape probabilities. In particular, ona directed graph with n vertices and m edges, we show how tocompute each quantity in time Õ(m[superscript 3/4]n + mn[superscript 2/3]), wherethe Õ notation suppresses polylog factors in n, the desired accuracy, and the appropriate condition number (i.e. themixing time or restart probability). Our result improves upon the previous fastest running times for these problems, previous results either invoke a general purpose linearsystem solver on a n × n matrix with m non-zero entries, or depend polynomially on the desired error or natural condition numberassociated with the problem (i.e. the mixing time or restart probability). For sparse graphs, we obtain a running time of Õ(n[superscript 7/4]), breaking the O(n[superscript 2]) barrier of the best running time one couldhope to achieve using fast matrix multiplication. We achieve our result by providing a similar running time improvementfor solving directed Laplacian systems, a natural directedor asymmetric analog of the well studied symmetric or undirected Laplaciansystems. We show how to solve such systems in time Õ(m[superscrip 3/4]n + mn[superscript 2/3]), and efficiently reduce a broad range of problems to solving Õ(1) directed Laplacian systems on Eulerian graphs. We hope these resultsand our analysis open the door for further study into directedspectral graph theory.National Science Foundation (U.S.) (Grant 1111109

Almost-linear-time algorithms for Markov chains and new spectral primitives for directed graphs

In this paper, we begin to address the longstanding algorithmic gap between general and reversible Markov chains. We develop directed analogues of several spectral graph-the oretic tools that had previously been available only in the undirected setting, and for which it was not clear that directed versions even existed. In particular, we provide a notion of approximation for directed graphs, prove sparsifiers under this notion always exist, and show how to construct them in almost linear time. Using this notion of approximation, we design the first almost-linear-time directed Laplacian system solver, and, by leveraging the recent framework of [Cohen-Kelner-Peebles-Peng-Sidford-Vladu, FOCS'16], we also obtain almost-linear-time algorithms for computing the stationary distribution of a Markov chain, computing expected commute times in a directed graph, and more. For each problem, our algorithms improve the previous best running times of O((nm [superscript 3/4] + n[superscript 2/3]m) log[superscript O(1)] (nkϵ[superscript -1])) to O((m + n2[superscript O (√log n loglogn))] log[superscript O(1)] (nkϵ [superscript -1])) where n is the number of vertices in the graph, m is the number of edges, κ is a natural condition number associated with the problem, and ϵ is the desired accuracy. We hope these results open the door for further studies into directed spectral graph theory, and that they will serve as a stepping stone for designing a new generation of fast algorithms for directed graphs