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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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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.
Routing in undirected graphs with constant congestion
 CoRR
"... Given an undirected graph G = (V, E), a collection (s1, t1),..., (sk, tk) of k demand pairs, and an integer c, the goal in the Edge Disjoint Paths with Congestion problem is to connect maximum possible number of the demand pairs by paths, so that the maximum load on any edge (called edge congestion) ..."
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Cited by 10 (5 self)
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Given an undirected graph G = (V, E), a collection (s1, t1),..., (sk, tk) of k demand pairs, and an integer c, the goal in the Edge Disjoint Paths with Congestion problem is to connect maximum possible number of the demand pairs by paths, so that the maximum load on any edge (called edge congestion) does not exceed c. We show an efficient randomized algorithm to route Ω(OPT / poly log k) demand pairs with congestion at most 14, where OPT is the maximum number of pairs that can be simultaneously routed on edgedisjoint paths. The best previous algorithm that routed Ω(OPT / poly log n) pairs required congestion poly(log log n), and for the setting where the maximum allowed congestion is bounded by a constant c, the best previous algorithms could only guarantee the routing of OPT/n O(1/c) pairs. We also introduce a new type of vertex sparsifiers that we call integral flow sparsifiers, that approximately preserve both fractional and integral routings, and show an algorithm to construct such sparsifiers.
On mimicking networks representing minimum terminal cuts
, 2012
"... Given a capacitated undirected graph G = (V,E) with a set of terminals K ⊂ V, a mimicking network is a smaller graph H = (VH, EH) that exactly preserves all the minimum cuts between the terminals. Specifically, the vertex set of the sparsifier VH contains the set of terminals K and for every biparti ..."
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Cited by 4 (1 self)
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Given a capacitated undirected graph G = (V,E) with a set of terminals K ⊂ V, a mimicking network is a smaller graph H = (VH, EH) that exactly preserves all the minimum cuts between the terminals. Specifically, the vertex set of the sparsifier VH contains the set of terminals K and for every bipartition U,K − U of the terminals K, the size of the minimum cut separating U from K − U in G is exactly equal to the size of the minimum cut separating U from K − U in H. This notion of a mimicking network was introduced by Hagerup, Katajainen, Nishimura and Ragde [HKNR95] who also exhibited a mimicking network of size 22 k for every graph with k terminals. The best known lower bound on the size of a mimicking network is linear in the number of terminals. More precisely, the best known lower bound is k + 1 for graphs with k terminals [CSWZ00]. In this work, we improve both the upper and lower bounds reducing the doublyexponential gap between them to a singleexponential gap. Specifically, we obtain the following upper and lower bounds on mimicking networks: • Given a graph G, we exhibit a construction of mimicking network with at most (K−1)’th Dedekind number ( ≈ 2 ( (k−1)b(k−1)/2c)) of vertices (independent of size of V). Furthermore, we show that the construction is optimal among all restricted mimicking networks – a natural class of mimicking networks that are obtained by clustering vertices together. • There exists graphs with k terminals that have no mimicking network of size smaller than 2 k−1 2. We also exhibit improved constructions of mimicking networks for trees and graphs of bounded treewidth.
Degree3 Treewidth Sparsifiers∗
, 2014
"... We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, V (H)  is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on nodedisj ..."
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Cited by 1 (1 self)
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We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, V (H)  is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on nodedisjoint paths, and computing minors seems easier in subcubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph G of treewidth k, computes a topological minor H of G such that (i) the treewidth of H is Ω(k/polylog(k)); (ii) V (H)  = O(k4); and (iii) the maximum vertex degree in H is 3. The running time of the algorithm is polynomial in V (G)  and k. Our result is in contrast to the known fact that unless NP ⊆ coNP/poly, treewidth does not admit polynomialsize kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves nodedisjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small goodquality vertexcut sparsifiers that are also minors of the original graph. 1
Degree3 Treewidth Sparsifiers
, 2014
"... We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, V (H)  is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on nodedisj ..."
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We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, V (H)  is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on nodedisjoint paths, and computing minors seems easier in subcubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph G of treewidth k, computes a topological minor H of G such that (i) the treewidth of H is Ω(k/polylog(k)); (ii) V (H)  = O(k4); and (iii) the maximum vertex degree in H is 3. The running time of the algorithm is polynomial in V (G)  and k. Our result is in contrast to the known fact that unless NP ⊆ coNP/poly, treewidth does not admit polynomialsize kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves nodedisjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small goodquality vertexcut sparsifiers that are also minors of the original graph.