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105,658
VERTEX SPARSIFICATION AND OBLIVIOUS REDUCTIONS
, 2009
"... Given an undirected, capacitated graph G = (V, E) and a set K ⊂ V of terminals of size k, we construct an undirected, capacitated graph G ′ = (K, E ′ ) for which the cutfunction approximates the value of every minimum cut separating any subset U of terminals from the remaining terminals K − U. We ..."
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Given an undirected, capacitated graph G = (V, E) and a set K ⊂ V of terminals of size k, we construct an undirected, capacitated graph G ′ = (K, E ′ ) for which the cutfunction approximates the value of every minimum cut separating any subset U of terminals from the remaining terminals K − U. We refer to this graph G ′ as a cutsparsifier, and we prove that there are cutsparsifiers that can approximate all these minimum cuts in G to within an approximation factor that only depends polylogarithmically on k, the number of terminals. We prove such cutsparsifiers exist through a zerosum game, and we construct such sparsifiers through oblivious routing guarantees. These results allow us to derive a more general theory of Steiner cut and flow problems, and allow us to obtain approximation algorithms with guarantees independent of the size of the graph for a number of graph partitioning, graph layout and multicommodity flow problems for which such guarantees were previously unknown.
Improved Sparsification
, 1993
"... In previous work we introduced sparsification, a technique that transforms fully dynamic algorithms for sparse graphs into ones that work on any graph, with a logarithmic increase in complexity. In this work we describe an improvement on this technique that avoids the logarithmic overhead. Using ..."
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Cited by 29 (5 self)
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. Using our improved sparsification technique, we keep track of the following properties: minimum spanning forest, best swap, connectivity, 2edgeconnectivity, and bipartiteness, in time O(n 1/2 ) per edge insertion or deletion; 2vertexconnectivity and 3vertexconnectivity, in time O(n) per
Proof verification and hardness of approximation problems
 IN PROC. 33RD ANN. IEEE SYMP. ON FOUND. OF COMP. SCI
, 1992
"... We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probabilit ..."
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Cited by 793 (39 self)
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vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige, Goldwasser, Lovász, Safra and Szegedy [42], and Arora and Safra [6] and shows that there exists a positive ɛ such that approximating
A new approach to the maximum flow problem
 JOURNAL OF THE ACM
, 1988
"... All previously known efficient maximumflow algorithms work by finding augmenting paths, either one path at a time (as in the original Ford and Fulkerson algorithm) or all shortestlength augmenting paths at once (using the layered network approach of Dinic). An alternative method based on the pre ..."
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Cited by 663 (33 self)
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to be shortest paths. The algorithm and its analysis are simple and intuitive, yet the algorithm runs as fast as any other known method on dense. graphs, achieving an O(n³) time bound on an nvertex graph. By incorporating the dynamic tree data structure of Sleator and Tarjan, we obtain a version
Vertex Sparsification and Universal Rounding Algorithms
, 2011
"... Suppose we are given a gigantic communication network, but are only interested in a small number of nodes (clients). There are many routing problems we could be asked to solve for our clients. Is there a much smaller network that we could write down on a sheet of paper and put in our pocket that a ..."
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Cited by 1 (0 self)
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 that approximately preserves all the relevant communication properties of the original network? As we will demonstrate, the answer to this question is YES, and we call this smaller network a vertex sparsifier. In fact, if we are asked to solve a sequence of optimization problems characterized by cuts or flows, we
Complexity of finding embeddings in a ktree
 SIAM JOURNAL OF DISCRETE MATHEMATICS
, 1987
"... A ktree is a graph that can be reduced to the kcomplete graph by a sequence of removals of a degree k vertex with completely connected neighbors. We address the problem of determining whether a graph is a partial graph of a ktree. This problem is motivated by the existence of polynomial time al ..."
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Cited by 380 (1 self)
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A ktree is a graph that can be reduced to the kcomplete graph by a sequence of removals of a degree k vertex with completely connected neighbors. We address the problem of determining whether a graph is a partial graph of a ktree. This problem is motivated by the existence of polynomial time
On sparsification for computing treewidth
 In Proceedings of IPEC
, 2013
"... Abstract. We investigate whether an nvertex instance (G, k) of Treewidth, asking whether the graph G has treewidth at most k, can efficiently be made sparse without changing its answer. By giving a special form of orcrosscomposition, we prove that this is unlikely: if there is an > 0 and a p ..."
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Cited by 3 (0 self)
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Abstract. We investigate whether an nvertex instance (G, k) of Treewidth, asking whether the graph G has treewidth at most k, can efficiently be made sparse without changing its answer. By giving a special form of orcrosscomposition, we prove that this is unlikely: if there is an > 0 and a
A Data Structure for Dynamic Trees
, 1983
"... A data structure is proposed to maintain a collection of vertexdisjoint trees under a sequence of two kinds of operations: a link operation that combines two trees into one by adding an edge, and a cut operation that divides one tree into two by deleting an edge. Each operation requires O(log n) ti ..."
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Cited by 343 (21 self)
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A data structure is proposed to maintain a collection of vertexdisjoint trees under a sequence of two kinds of operations: a link operation that combines two trees into one by adding an edge, and a cut operation that divides one tree into two by deleting an edge. Each operation requires O(log n
Spectral Sparsification and Spectrally Thin Trees
, 2012
"... We provide results of intensive experimental data in order to investigate the existence of spectrally thin trees and unweighted spectral sparsifiers for graphs with small expansion. In addition, we also survey and prove some partial results on the existence of spectrally thin trees on dense graphs w ..."
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We provide results of intensive experimental data in order to investigate the existence of spectrally thin trees and unweighted spectral sparsifiers for graphs with small expansion. In addition, we also survey and prove some partial results on the existence of spectrally thin trees on dense graphs
Extensions and Limits to Vertex Sparsification
"... Suppose we are given a graph G = (V, E) and a set of terminals K ⊂ V. We consider the problem of constructing a graph H = (K, EH) that approximately preserves the congestion of every multicommodity flow with endpoints supported in K. We refer to such a graph as a flow sparsifier. We prove that there ..."
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Cited by 11 (2 self)
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derived using vertex sparsification in [22]. We also prove an Ω(loglogk) lower bound for how well a flow sparsifier can simultaneously approximate the congestion of every multicommodity flow in the original graph. Our proof crucially relies on a geometric phenomenon pertaining to the unit congestion
Results 1  10
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105,658