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18
Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs
, 2010
"... We introduce a new approach to computing an approximately maximum st flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a system of linear equations in a Laplacian matrix, and thus may be ..."
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Cited by 40 (5 self)
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We introduce a new approach to computing an approximately maximum st flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a system of linear equations in a Laplacian matrix, and thus may be approximately computed in nearlylinear time. Using this approach, we develop the fastest known algorithm for computing approximately maximum st flows. For a graph having n vertices and m edges, our algorithm computes a (1−ɛ)approximately maximum st flow in time 1 Õ ( mn 1/3 ɛ −11/3). A dual version of our approach computes a (1 + ɛ)approximately minimum st cut in time Õ ( m + n 4/3 ɛ −16/3) , which is the fastest known algorithm for this problem as well. Previously, the best dependence on m and n was achieved by the algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute approximately maximum st flows in time Õ ( m √ nɛ −1) , and approximately minimum st cuts in time Õ ( m + n 3/2 ɛ −3). Research partially supported by NSF grant CCF0843915.
A General Framework for Graph Sparsification
, 2011
"... We present a general framework for constructing cut sparsifiers in undirected graphs — weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of (1 ± ǫ). Using this framework, we simplify, unify and improve upon previous sparsification results ..."
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Cited by 21 (1 self)
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We present a general framework for constructing cut sparsifiers in undirected graphs — weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of (1 ± ǫ). Using this framework, we simplify, unify and improve upon previous sparsification results. As simple instantiations of this framework, we show that sparsifiers can be constructed by sampling edges according to their strength (a result of Benczúr and Karger), effective resistance (a result of Spielman and Srivastava), edge connectivity, or by sampling random spanning trees. Sampling according to edge connectivity is the most aggressive method, and the most challenging to analyze. Our proof that this method produces sparsifiers resolves an open question of Benczúr and Karger. While the above results are interesting from a combinatorial standpoint, we also prove new algorithmic results. In particular, we develop techniques that give the first (optimal) O(m)time sparsification algorithm for unweighted graphs. Our algorithm has a running time of O(m) + Õ(n/ǫ²) for weighted graphs, which is also linear unless the input graph is very sparse itself. In both cases, this improves upon the previous best running times (due to Benczúr and Karger) of O(m log² n) (for the unweighted case) and O(m log³ n) (for the weighted case) respectively. Our algorithm constructs sparsifiers that contain O(n log n/ǫ²) edges in expectation; the only known construction of sparsifiers with fewer edges is by a substantially slower algorithm running in O(n 3 m/ǫ 2) time. A key ingredient of our proofs is a natural generalization of Karger’s bound on the number of small cuts in an undirected graph. Given the numerous applications of Karger’s bound, we suspect that our generalization will also be of independent interest.
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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Cited by 14 (7 self)
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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 Maximum Flows in Nearly Linear Time
 In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science
, 2013
"... We introduce a new approach to the maximum flow problem in undirected, capacitated graphs using αcongestionapproximators: easytocompute functions that approximate the congestion required to route singlecommodity demands in a graph to within a factor of α. Our algorithm maintains an arbitrary fl ..."
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Cited by 12 (0 self)
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We introduce a new approach to the maximum flow problem in undirected, capacitated graphs using αcongestionapproximators: easytocompute functions that approximate the congestion required to route singlecommodity demands in a graph to within a factor of α. Our algorithm maintains an arbitrary flow that may have some residual excess and deficits, while taking steps to minimize a potential function measuring the congestion of the current flow plus an overestimate of the congestion required to route the residual demand. Since the residual term overestimates, the descent process gradually moves the contribution to our potential function from the residual term to the congestion term, eventually achieving a flow routing the desired demands with nearly minimal congestion after Õ(αε−2 log2 n) iterations. Our approach is similar in spirit to that used by Spielman and Teng (STOC 2004) for solving Laplacian systems, and we summarize our approach as trying to do for `∞flows what they do for `2flows. Together with a nearly linear time construction of a no(1)congestionapproximator, we obtain 1 + εoptimal singlecommodity flows undirected graphs in time m1+o(1)ε−2, yielding the fastest known algorithm for that problem. Our requirements of a congestionapproximator are quite low, suggesting even faster and simpler algorithms for certain classes of graphs. For example, an αcompetitive oblivious routing tree meets our definition, even without knowing how to route the tree back in the graph. For graphs of conductance φ, a trivial φ−1congestionapproximator gives an extremely simple algorithm for finding 1 + εoptimalflows in time Õ(mφ−1). 1
Navigating Central Path with Electrical Flows: From Flows to Matchings, and Back
 FOCS
, 2013
"... We present an Õ(m ..."
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Approximating the exponential, the Lanczos method, and an Õ(m)time spectral algorithm for balanced separator
 IN: PROCEEDINGS OF THE 44TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 2012
"... We give a novel spectral approximation algorithm for the balanced separator problem that, given a graph G, a constant balance b ∈ (0, 1/2], and a parameter γ, either finds an Ω(b)balanced cut of conductance O ( √ γ) in G, or outputs a certificate that all bbalanced cuts in G have conductance at le ..."
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Cited by 10 (3 self)
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We give a novel spectral approximation algorithm for the balanced separator problem that, given a graph G, a constant balance b ∈ (0, 1/2], and a parameter γ, either finds an Ω(b)balanced cut of conductance O ( √ γ) in G, or outputs a certificate that all bbalanced cuts in G have conductance at least γ, and runs in time Õ(m). This settles the question of designing asymptotically optimal spectral algorithms for balanced separator. Our algorithm relies on a variant of the heat kernel random walk and requires, as a subroutine, an algorithm to compute exp(−L)v where L is the Laplacian of a graph related to G and v is a vector. Algorithms for computing the matrixexponentialvector product efficiently comprise our next set of results. Our main result here is a new algorithm which computes a good approximation to exp(−A)v for a class of symmetric positive semidefinite (PSD) matrices A and a given vector u, in time roughly Õ(m A), where m A is the number of nonzero entries of A. This uses, in a nontrivial way, the breakthrough result of Spielman and Teng on inverting symmetric and diagonallydominant matrices in Õ(m A) time. Finally, we prove that e −x can be uniformly approximated up to a small additive error, in a nonnegative interval [a, b] with a polynomial of
Balanced Partitions of Trees and Applications
, 2012
"... We study the kBALANCED PARTITIONING problem in which the vertices of a graph are to be partitioned into k sets of size at most ⌈n/k ⌉ while minimising the cut size, which is the number of edges connecting vertices in different sets. The problem is well studied for general graphs, for which it canno ..."
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Cited by 9 (3 self)
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We study the kBALANCED PARTITIONING problem in which the vertices of a graph are to be partitioned into k sets of size at most ⌈n/k ⌉ while minimising the cut size, which is the number of edges connecting vertices in different sets. The problem is well studied for general graphs, for which it cannot be approximated within any factor in polynomial time. However, little is known about restricted graph classes. We show that for trees kBALANCED PARTITIONING remains surprisingly hard. In particular, approximating the cut size is APXhard even if the maximum degree of the tree is constant. If instead the diameter of the tree is bounded by a constant, we show that it is NPhard to approximate the cut size within nc, for any constant c < 1. In the face of the hardness results, we show that allowing nearbalanced solutions, in which there are at most (1+ε)⌈n/k ⌉ vertices in any of the k sets, admits a PTAS for trees. Remarkably, the computed cut size is no larger than that of an optimal balanced solution. In the final section of our paper, we harness results on embedding graph metrics into tree metrics to extend our PTAS for trees to general graphs. In addition to being conceptually simpler and easier to analyse, our scheme improves the best factor known on the cut size of nearbalanced solutions from O(log 1.5 (n)/ε 2) [Andreev and Räcke TCS 2006] to O(log n), for weighted graphs. This also settles a question posed by Andreev and Räcke of whether an algorithm with approximation guarantees on the cut size independent from ε exists.
Fast Approximation Algorithms for Graph Partitioning Using Spectral and SemidefiniteProgramming Techniques
, 2011
"... Graphpartitioning problems are a central topic of research in the study of approximation algorithms. They are of interest to both theoreticians, for their farreaching connections to different areas of mathematics, and to practitioners, as algorithms for graph partitioning can be used as fundamenta ..."
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Cited by 3 (1 self)
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Graphpartitioning problems are a central topic of research in the study of approximation algorithms. They are of interest to both theoreticians, for their farreaching connections to different areas of mathematics, and to practitioners, as algorithms for graph partitioning can be used as fundamental building blocks in many applications, such as image segmentation and clustering. While many theoretical approximation algorithms exist for graph partitioning, they often rely on multicommodityflow computations that run in quadratic time in the worst case and are too timeconsuming for the massive graphs that are prevalent in today’s applications. In this dissertation, we study the design of approximation algorithms that yield strong approximation guarantees, while running in subquadratic time and relying on computational procedures that are often fast in practice. The results that we describe encompass two different approaches to the construction of such fast algorithms. Our first result exploits the CutMatching game of Khandekar, Rao and Vazirani [41], an elegant framework for designing graphpartitioning algorithms that rely on singlecommodity, rather than multicommodity, maximum flow. Within this framework, we give two novel algorithms that achieve an O(log n)approximation for the problem of finding the cut of minimum