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An almostlineartime algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations
"... In this paper we present an almost linear time algorithm for solving approximate maximum flow in undirected graphs. In particular, given a graph with m edges we show how to produce a 1−ε approximate maximum flow in time O(m 1+o(1) · ε −2). Furthermore, we present this algorithm as part of a general ..."
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In this paper we present an almost linear time algorithm for solving approximate maximum flow in undirected graphs. In particular, given a graph with m edges we show how to produce a 1−ε approximate maximum flow in time O(m 1+o(1) · ε −2). Furthermore, we present this algorithm as part of a general framework that also allows us to achieve a running time of O(m 1+o(1) ε −2 k 2) for the maximum concurrent kcommodity flow problem, the first such algorithm with an almost linear dependence on m. We also note that independently Jonah Sherman has produced an almost linear time algorithm for maximum flow and we thank him for coordinating submissions.
NEARLY LINEAR TIME ALGORITHMS FOR PRECONDITIONING AND SOLVING SYMMETRIC, DIAGONALLY DOMINANT LINEAR SYSTEMS
, 2014
"... We present a randomized algorithm that on input a symmetric, weakly diagonally dominant nbyn matrix A with m nonzero entries and an nvector b produces an x ̃ such that ‖x ̃ − A†b‖A ≤ ‖A†b‖A in expected time O(m logc n log(1/)) for some constant c. By applying this algorithm inside the inverse p ..."
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We present a randomized algorithm that on input a symmetric, weakly diagonally dominant nbyn matrix A with m nonzero entries and an nvector b produces an x ̃ such that ‖x ̃ − A†b‖A ≤ ‖A†b‖A in expected time O(m logc n log(1/)) for some constant c. By applying this algorithm inside the inverse power method, we compute approximate Fiedler vectors in a similar amount of time. The algorithm applies subgraph preconditioners in a recursive fashion. These preconditioners improve upon the subgraph preconditioners first introduced by Vaidya in 1990. For any symmetric, weakly diagonally dominant matrix A with nonpositive offdiagonal entries and k ≥ 1, we construct in time O(m logc n) a preconditioner B of A with at most 2(n − 1) +O((m/k) log39 n) nonzero offdiagonal entries such that the finite generalized condition number κf (A,B) is at most k, for some other constant c. In the special case when the nonzero structure of the matrix is planar the corresponding linear system solver runs in expected time O(n log2 n+n logn log logn log(1/)). We hope that our introduction of algorithms of low asymptotic complexity will lead to the development of algorithms that are also fast in practice.
Analyzing Massive Graphs in the Semistreaming Model
"... Massive graphs arise in a many scenarios, for example, traffic data analysis in large networks, large scale scientific experiments, and clustering of large data sets. The semistreaming model was proposed for processing massive graphs. In the semistreaming model, we have a random accessible memory ..."
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Massive graphs arise in a many scenarios, for example, traffic data analysis in large networks, large scale scientific experiments, and clustering of large data sets. The semistreaming model was proposed for processing massive graphs. In the semistreaming model, we have a random accessible memory which is nearlinear in the number of vertices. The input graph (or equivalently, edges in the graph) is presented as a sequential list of edges (insertiononly model) or edge insertions and deletions (dynamic model). The list is readonly but we may make multiple passes over the list. There has been a few results in the insertiononly model such as computing distance spanners and approximating the maximum matching. In this thesis, we present some algorithms and techniques for (i) solving more complex problems in the semistreaming model, (for example, problems in the dynamic model) and (ii) having better solutions for the problems which have been studied (for example, the maximum matching problem). In course of both of these, we develop new techniques with broad applications and explore the rich tradeoffs between the complexity of models (insertiononly streams vs. dynamic streams), the number of passes, space, accuracy, and running time. 1. We initiate the study of dynamic graph streams. We start with basic problems such as the connectivity problem and computing the minimum spanning tree. These problems are This dissertation is available at ScholarlyCommons:
Near Linear Time Approximation Schemes for Uncapacitated and Capacitated b–Matching Problems in Nonbipartite Graphs
, 2013
"... We present the first fully polynomial approximation schemes for the maximum weighted (uncapacitated or capacitated) b–Matching problem for nonbipartite graphs that run in time (near) linear in the number of edges, that is, given any δ> 0 the algorithm produces a (1 − δ) approximation in O(m pol ..."
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We present the first fully polynomial approximation schemes for the maximum weighted (uncapacitated or capacitated) b–Matching problem for nonbipartite graphs that run in time (near) linear in the number of edges, that is, given any δ> 0 the algorithm produces a (1 − δ) approximation in O(m poly(δ−1, logn)) time. We provide fractional solutions for the standard linear programming formulations for these problems and subsequently also provide fully polynomial (near) linear time approximation schemes for rounding the fractional solutions. Through these problems as a vehicle, we also present several ideas in the context of solving linear programs approximately using fast primaldual algorithms. First, we show that approximation algorithms can be used to reduce the width of the formulation, and as a consequence we induce faster convergence. Second, even though the dual of these problems have exponentially many variables and an efficient exact computation of dual weights is infeasible, we can efficiently compute and use a sparse approximation of the dual weights using a combination of (i) adding perturbation to the constraints of the polytope and (ii) amplification followed by thresholding of the dual weights. These algorithms also have the advantage that they use O(n poly(δ−1, logn)) storage space and only make O(δ−4 log2(1/δ) logn) (or better) passes over a read only list of edges. These algorithms therefore can be run in the semistreaming model and serve as exemplars where algorithms and ideas developed for the streaming model gives us algorithms for combinatorial optimization problems that were not known in absence of the streaming constraints.
Smaller Steps for Faster Algorithms: A New Approach to Solving Linear Systems
, 2013
"... In this thesis we study iterative algorithms with simple sublinear time update steps, and we show how a mix of of data structures, randomization, and results from numerical analysis allow us to achieve faster algorithms for solving linear systems in a variety of different regimes. First we present ..."
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In this thesis we study iterative algorithms with simple sublinear time update steps, and we show how a mix of of data structures, randomization, and results from numerical analysis allow us to achieve faster algorithms for solving linear systems in a variety of different regimes. First we present a simple combinatorial algorithm for solving symmetric diagonally dominant (SDD) systems of equations that improves upon the best previously known running time for solving such system in the standard unitcost RAM model. Then we provide a general method for convex optimization that improves this simple algorithm's running time as special case. Our results include the following: * We achieve the best known running time of 0 (m log312 1 log log n log(e 1 log n)) for solving Symmetric Diagonally Dominant (SDD) system of equations in the standard unitcost RAM model. e We obtain a faster asymptotic running time than conjugate gradient for solving a broad class of symmetric positive definite systems of equations.