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164
On the SingleSource Unsplittable Flow Problem
, 1998
"... Let G = (V; E) be a capacitated directed graph with a source s and k terminals t i with demands d i , 1 i k. We would like to concurrently route every demand on a single path from s to the corresponding terminal without violating the capacities. There are several interesting and important varia ..."
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Cited by 48 (2 self)
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Let G = (V; E) be a capacitated directed graph with a source s and k terminals t i with demands d i , 1 i k. We would like to concurrently route every demand on a single path from s to the corresponding terminal without violating the capacities. There are several interesting and important variations of this unsplittable flow problem. If the
The AllorNothing Multicommodity Flow Problem
 IN PROCEEDINGS OF THE 36TH ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC)
, 2004
"... ..., the same as that for edp [10]. Our algorithm extends to the case where each pair siti has a demand di associated with it and we need to completely route di to get credit for pair i. We also consider the online admission control version where pairs arrive online and the algorithm has to decide i ..."
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Cited by 42 (13 self)
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..., the same as that for edp [10]. Our algorithm extends to the case where each pair siti has a demand di associated with it and we need to completely route di to get credit for pair i. We also consider the online admission control version where pairs arrive online and the algorithm has to decide immediately on its arrival whether to accept it or not. We obtain a randomized algorithm with a competitive ratio that is similar to the approximation ratio for the offline algorithm.
Edge Disjoint Paths Revisited
 In Proceedings of the 14th ACMSIAM Symposium on Discrete Algorithms
, 2003
"... The approximability of the maximum edge disjoint paths problem (EDP) in directed graphs was seemingly settled by the )hardness result of Guruswami et al. [10] and the O( m) approximation achievable via both the natural LP relaxation [19] and the greedy algorithm [11, 12]. Here m is the numb ..."
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Cited by 40 (4 self)
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The approximability of the maximum edge disjoint paths problem (EDP) in directed graphs was seemingly settled by the )hardness result of Guruswami et al. [10] and the O( m) approximation achievable via both the natural LP relaxation [19] and the greedy algorithm [11, 12]. Here m is the number of edges in the graph. However, we observe that the hardness of approximation shown in [10] applies to sparse graphs and hence when expressed as a function of n, the number of vertices, only an \Omega\Gamma n )hardness follows. On the other hand, the O( m)approximation algorithms do not guarantee a sublinear (in terms of n) approximation algorithm for dense graphs. We note that a similar gap exists in the known results on the integrality gap of the natural LP relaxation: an \Omega\Gamma n) lower bound and an O( m) upper bound. Motivated by this discrepancy in the upper and lower bounds we study algorithms for the EDP in directed and undirected graphs obtaining improved approximation ratios. We show that the greedy algorithm has an approximation ratio of O(min(n m)) in undirected graphs and a ratio of O(min(n m)) in directed graphs. For ayclic graphs we give an O( n log n) approximation via LP rounding. These are the first sublinear approximation ratios for EDP. Our results also extend to EDP with weights and to the unsplittable flow problem with uniform edge capacities.
New Algorithmic Aspects Of The Local Lemma With Applications To Routing And Partitioning
"... . The Lov'asz Local Lemma (LLL) is a powerful tool that is increasingly playing a valuable role in computer science. The original lemma was nonconstructive; a breakthrough of Beck and its generalizations (due to Alon and Molloy & Reed) have led to constructive versions. However, these metho ..."
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Cited by 31 (6 self)
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. The Lov'asz Local Lemma (LLL) is a powerful tool that is increasingly playing a valuable role in computer science. The original lemma was nonconstructive; a breakthrough of Beck and its generalizations (due to Alon and Molloy & Reed) have led to constructive versions. However, these methods do not capture some classes of applications of the LLL. We make progress on this, by providing algorithmic approaches to two families of applications of the LLL. The first provides constructive versions of certain applications of an extension of the LLL (modeling, e.g., hypergraphpartitioning and lowcongestion routing problems); the second provides new algorithmic results on constructing disjoint paths in graphs. Our results can also be seen as constructive upper bounds on the integrality gap of certain packing problems. One common theme of our work is a "gradual rounding" approach.
On the kSplittable Flow Problem
, 2002
"... In traditional multicommodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However, ..."
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Cited by 31 (3 self)
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In traditional multicommodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However,
OnLine Randomized Call Control Revisited
 SIAM J. COMPUTING
, 2001
"... We consider the problem of online call admission and routing on trees and meshes. Previous work gave randomized online algorithms for these problems and proved that they have optimal (up to constant factors) competitive ratios. However, these algorithms can obtain very low profit with high probabi ..."
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Cited by 29 (5 self)
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We consider the problem of online call admission and routing on trees and meshes. Previous work gave randomized online algorithms for these problems and proved that they have optimal (up to constant factors) competitive ratios. However, these algorithms can obtain very low profit with high probability. We investigate the question of devising for these problems online competitive algorithms that also guarantee a "good" solution with "good" probability. We give a new
Graph decomposition and a greedy algorithm for edgedisjoint paths
 In Proceedings of the 15th ACMSIAM Symposium on Discrete Algorithms (SODA
, 2004
"... Abstract Given a directed graph G = (V, E) with n vertices and a parameter l> = 1, we present an algorithm that finds a cut (set of edges) of size O((n2/l2)log2(n/l)) whose removal separates every pair of vertices (s,t) in G such that the minimum distance between s and t in G is at least l. This ..."
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Cited by 28 (0 self)
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Abstract Given a directed graph G = (V, E) with n vertices and a parameter l> = 1, we present an algorithm that finds a cut (set of edges) of size O((n2/l2)log2(n/l)) whose removal separates every pair of vertices (s,t) in G such that the minimum distance between s and t in G is at least l. This theorem implies a nearly tight analysis of the greedy algorithm for finding edgedisjoint paths in directed graphs, and gives the best known approximation factor for this problem in terms of the number of vertices.
B.: A quasiPTAS for unsplittable flow on line graphs
 In: STOC
, 2006
"... We study the Unsplittable Flow Problem (UFP) on a line graph, focusing on the longstanding open question of whether the problem is APXhard. We describe a deterministic quasipolynomial time approximation scheme for UFP on line graphs, thereby ruling out an APXhardness result, unless NP ⊆ DTIME(2 ..."
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Cited by 28 (3 self)
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We study the Unsplittable Flow Problem (UFP) on a line graph, focusing on the longstanding open question of whether the problem is APXhard. We describe a deterministic quasipolynomial time approximation scheme for UFP on line graphs, thereby ruling out an APXhardness result, unless NP ⊆ DTIME(2polylog(n)). Our result requires a quasipolynomial bound on all edge capacities and demands in the input instance. Earlier results on this problem included a polynomial time (2+ ε)approximation under the assumption that no demand exceeds any edge capacity (the “nobottleneck assumption”) and a superconstant integrality gap if this assumption did not hold. Unlike most earlier work on UFP, our results do not require a nobottleneck assumption.
An O( √ n) approximation and integrality gap for disjoint paths and unsplittable flow
 Theory of Computing
, 2006
"... Abstract: We consider the maximization version of the edgedisjoint path problem (EDP). In undirected graphs and directed acyclic graphs, we obtain an O ( √ n) upper bound on the approximation ratio where n is the number of nodes in the graph. We show this by establishing the upper bound on the int ..."
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Cited by 27 (3 self)
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Abstract: We consider the maximization version of the edgedisjoint path problem (EDP). In undirected graphs and directed acyclic graphs, we obtain an O ( √ n) upper bound on the approximation ratio where n is the number of nodes in the graph. We show this by establishing the upper bound on the integrality gap of the natural relaxation based on multicommodity flows. Our upper bound matches within a constant factor a lower bound of Ω ( √ n) that is known for both undirected and directed acyclic graphs. The best previous upper bounds on the integrality gaps were O(min{n 2/3, √ m}) for undirected graphs and O(min { √ nlogn, √ m}) for directed acyclic graphs; here m is the number of edges in the graph. These bounds are also the best known approximation ratios for these problems. Our bound also extends to the unsplittable flow problem (UFP) when the maximum demand is at most the minimum capacity. ACM Classification: C.2.0, F.2.2, G.1.6, G.3 AMS Classification: 68W20, 68W25, 90C59