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37
On the number of iterations for dantzigwolfe optimization and packingcovering approximation algorithms
 In Proceedings of the 7th International IPCO Conference
, 1999
"... We start with definitions given by Plotkin, Shmoys, and Tardos [16]. Given A ∈ IR m×n, b ∈ IR m and a polytope P ⊆ IR n,thefractional packing problem is to find an x ∈ P such that Ax ≤ b if such an x exists. An ɛapproximate solution to this problem is an x ∈ P such that Ax ≤ (1 + ɛ)b. Anɛrelaxed d ..."
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Cited by 23 (2 self)
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We start with definitions given by Plotkin, Shmoys, and Tardos [16]. Given A ∈ IR m×n, b ∈ IR m and a polytope P ⊆ IR n,thefractional packing problem is to find an x ∈ P such that Ax ≤ b if such an x exists. An ɛapproximate solution to this problem is an x ∈ P such that Ax ≤ (1 + ɛ)b. Anɛrelaxed decision
Using petaldecompositions to build a low stretch spanning tree
 in Proceedings of ACM STOC
, 2012
"... We prove that any graph G = (V,E) with n points and m edges has a spanning tree T such that∑ (u,v)∈E(G) dT (u, v) = O(m log n log log n). Moreover such a tree can be found in timeO(m log n log log n). Our result is obtained using a new petaldecomposition approach which guarantees that the radius o ..."
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Cited by 18 (1 self)
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We prove that any graph G = (V,E) with n points and m edges has a spanning tree T such that∑ (u,v)∈E(G) dT (u, v) = O(m log n log log n). Moreover such a tree can be found in timeO(m log n log log n). Our result is obtained using a new petaldecomposition approach which guarantees that the radius of each cluster in the tree is at most 4 times the radius of the induced subgraph of the cluster in the original graph.
Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs
, 2011
"... We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately t ..."
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Cited by 17 (0 self)
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We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately the same value as the original graph’s. In this new graph, for example, we can run the Õ(m3/2)time maximum flow algorithm of Goldberg and Rao to find an s– t minimum cut in Õ(n3/2) time. This corresponds to a (1 + ɛ)times minimum s–t cut in the original graph. A related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in Õ(m √ n) time. Our algorithm is also used to improve the running time of sparsest cut algorithms from Õ(mn) to Õ(n²). Our approach also accelerates several other recent cut and flow algorithms. Our algorithms are based on a general theorem analyzing the concentration of cut values near their expectation in random graphs.
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.
Navigating Central Path with Electrical Flows: From Flows to Matchings, and Back
 FOCS
, 2013
"... We present an Õ(m ..."
A New Approach to Computing Maximum Flows using Electrical Flows
 Proceedings of the 45th symposium on Theory of Computing  STOC ’13
, 2013
"... We give an algorithm which computes a (1−ɛ)approximately maximum st−flow in an undirected uncapacitated graph in time O ( 1 m ɛ F ·m log2 n) where F is the flow value. By trading this off against the KargerLevine algorithm for undirected graphs which takes Õ(m + nF) time, we obtain a running time ..."
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Cited by 10 (4 self)
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We give an algorithm which computes a (1−ɛ)approximately maximum st−flow in an undirected uncapacitated graph in time O ( 1 m ɛ F ·m log2 n) where F is the flow value. By trading this off against the KargerLevine algorithm for undirected graphs which takes Õ(m + nF) time, we obtain a running time of Õ(mn1/3 /ɛ 2/3) for uncapacitated graphs, improving the previous best dependence on ɛ by a factor of O(1/ɛ 3). Like the algorithm of Christiano, Kelner, Madry, Spielman and Teng, our algorithm reduces the problem to electrical flow computations which are carried out in linear time using fast Laplacian solvers. However, in contrast to previous work, our algorithm does not reweight the edges of the graph in any way, and instead uses local (i.e., non s − t) electrical flows to reroute the flow on congested edges. The algorithm is simple and may be viewed as trying to find a point at the intersection of two convex sets (the affine subspace of stflows of value F and the ℓ ∞ ball) by an accelerated version of the method of alternating projections due to Nesterov. By combining this with Ford and Fulkerson’s augmenting paths algorithm, we obtain an exact algorithm with running time Õ(m5/4F 1/4) for uncapacitated undirected graphs, improving the previous best running time of Õ(m+min(nF, m3/2)). We give a related algorithm with the same running time for approximate minimum cut, based on minimizing a smoothed version of the ℓ1 norm inside the cut space of the input graph. We show that the minimizer of this norm is related to an approximate blocking flow and use this to give an algorithm for computing a length k approximately blocking flow in time Õ(m√k).
Physarum Can Compute Shortest Paths
, 2012
"... Physarum Polycephalum is a slime mold that apparently is able to solve shortest path problems. A mathematical model has been proposed by biologists to describe the feedback mechanism used by the slime mold to adapt its tubular channelswhileforaging twofood sourcess0 ands1. Weprove that, under this m ..."
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Cited by 10 (0 self)
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Physarum Polycephalum is a slime mold that apparently is able to solve shortest path problems. A mathematical model has been proposed by biologists to describe the feedback mechanism used by the slime mold to adapt its tubular channelswhileforaging twofood sourcess0 ands1. Weprove that, under this model, the mass of the mold will eventually converge to the shortest s0s1 path of the network that the mold lies on, independently of the structure of the network or of the initial mass distribution. This matches the experimental observations by the biologists and can be seen as an example of a “natural algorithm”, that is, an algorithm developed by evolution over millions of years.
A nearlym logn time solver for SDD linear systems
 In Proceedings of the IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS
, 2011
"... ar ..."
A fast solver for a class of linear systems
, 2008
"... The solution of linear systems is a problem of fundamental theoretical importance but also one with a myriad of applications in numerical mathematics, engineering and science. Linear systems that are generated by realworld applications frequently fall into special classes. Recent research led to a ..."
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Cited by 8 (3 self)
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The solution of linear systems is a problem of fundamental theoretical importance but also one with a myriad of applications in numerical mathematics, engineering and science. Linear systems that are generated by realworld applications frequently fall into special classes. Recent research led to a fast algorithm for solving symmetric diagonally dominant (SDD) linear systems. We give an overview of this solver and survey the underlying notions and tools from algebra, probability and graph algorithms. We also discuss some of the many and diverse applications of SDD solvers.
Spectral Sparsification of Graphs: Theory and Algorithms
, 2013
"... Graph sparsification is the approximation of an arbitrary graph by a sparse graph. We explain what it means for one graph to be a spectral approximation of another and review the development of algorithms for spectral sparsification. In addition to being an interesting concept, spectral sparsificati ..."
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Cited by 7 (0 self)
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Graph sparsification is the approximation of an arbitrary graph by a sparse graph. We explain what it means for one graph to be a spectral approximation of another and review the development of algorithms for spectral sparsification. In addition to being an interesting concept, spectral sparsification has been an important tool in the design of nearly lineartime algorithms for solving systems of linear equations in symmetric, diagonally dominant matrices. The fast solution of these linear systems has already led to breakthrough results in combinatorial optimization, including a faster algorithm for finding approximate maximum flows and minimum cuts in an undirected network.