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45
Approximation Algorithms for Disjoint Paths Problems
, 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for w ..."
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Cited by 166 (0 self)
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The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in highspeed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.
Nearoptimal hardness results and approximation algorithms for edgedisjoint paths and related problems
 Journal of Computer and System Sciences
, 1999
"... We study the approximability of edgedisjoint paths and related problems. In the edgedisjoint paths problem (EDP), we are given a network G with sourcesink pairs (si, ti), 1 ≤ i ≤ k, and the goal is to find a largest subset of sourcesink pairs that can be simultaneously connected in an edgedisjo ..."
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Cited by 106 (11 self)
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We study the approximability of edgedisjoint paths and related problems. In the edgedisjoint paths problem (EDP), we are given a network G with sourcesink pairs (si, ti), 1 ≤ i ≤ k, and the goal is to find a largest subset of sourcesink pairs that can be simultaneously connected in an edgedisjoint manner. We show that in directed networks, for any ɛ> 0, EDP is NPhard to approximate within m 1/2−ɛ. We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any ɛ> 0, bounded length EDP is hard to approximate within m 1/2−ɛ even in undirected networks, and give an O ( √ m)approximation algorithm for it. For directed networks, we show that even the single sourcesink pair case (i.e. find the maximum number of paths of bounded length between a given sourcesink pair) is hard to approximate within m 1/2−ɛ, for any ɛ> 0.
SingleSource Unsplittable Flow
 In Proceedings of the 37th Annual Symposium on Foundations of Computer Science
, 1996
"... The maxflow mincut theorem of Ford and Fulkerson is based on an even more foundational result, namely Menger's theorem on graph connectivity. Menger's theorem provides a good characterization for the following singlesource disjoint paths problem: given a graph G, with a source vertex s ..."
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Cited by 61 (2 self)
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The maxflow mincut theorem of Ford and Fulkerson is based on an even more foundational result, namely Menger's theorem on graph connectivity. Menger's theorem provides a good characterization for the following singlesource disjoint paths problem: given a graph G, with a source vertex s and terminals t 1 , ..., t k , decide whether there exist edgedisjoint st i paths, for i = 1, ..., k. We consider a natural, NPhard generalization of this problem, which we call the singlesource unsplittable flow problem. We are given a source and terminals as before; but now each terminal t i has a demand ae i 1, and each edge e of G has a capacity c e 1. The problem is to decide whether one can choose a single st i path, for each i, so that the resulting set of paths respects the capacity constraints  the total amount of demand routed across any edge e must be bounded by the capacity c e . The main results of this paper are constantfactor approximation algorithms for three n...
Strongly Polynomial Algorithms for the Unsplittable Flow Problem
 In Proceedings of the 8th Conference on Integer Programming and Combinatorial Optimization (IPCO
, 2001
"... We provide the first strongly polynomial algorithms with the best approximation ratio for all three variants of the unsplittable ow problem (UFP). In this problem we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand an ..."
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Cited by 48 (1 self)
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We provide the first strongly polynomial algorithms with the best approximation ratio for all three variants of the unsplittable ow problem (UFP). In this problem we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path as to maximize the total profit of the satisfied terminal pairs subject to the capacity constraints. Classical UFP, in which demands must be lower than edge capacities, is known to have an O( m) approximation algorithm. We provide the same result with a strongly polynomial combinatorial algorithm. The extended UFP case is when some demands might be higher than edge capacities. For that case we both improve the current best approximation ratio and use strongly polynomial algorithms.
Improved approximation algorithms for unsplittable flow problems (Extended Abstract)
 In Proceedings of the 38th Annual Symposium on Foundations of Computer Science
, 1997
"... ) Stavros G. Kolliopoulos 1 Clifford Stein 1 Abstract In the singlesource unsplittable flow problem we are given a graph G; a source vertex s and a set of sinks t 1 ; : : : ; t k with associated demands. We seek a single st i flow path for each commodity i so that the demands are satisfied and ..."
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Cited by 45 (2 self)
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) Stavros G. Kolliopoulos 1 Clifford Stein 1 Abstract In the singlesource unsplittable flow problem we are given a graph G; a source vertex s and a set of sinks t 1 ; : : : ; t k with associated demands. We seek a single st i flow path for each commodity i so that the demands are satisfied and the total flow routed across any edge e is bounded by its capacity c e : The problem is an NPhard variant of max flow and a generalization of singlesource edgedisjoint paths with applications to scheduling, load balancing and virtualcircuit routing problems. In a significant development, Kleinberg gave recently constantfactor approximation algorithms for several natural optimization versions of the problem [18]. In this paper we give a generic framework that yields simpler algorithms and significant improvements upon the constant factors. Our framework, with appropriate subroutines, applies to all optimization versions previously considered and treats in a unified manner directed and u...
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 43 (11 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 39 (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.
Approximating DisjointPath Problems Using Packing Integer Programs
, 1998
"... In a packing integer program, we are given a matrix A and column vectors b; c with nonnegative entries. We seek a vector x of nonnegative integers, which maximizes c^T x; subject to Ax ≤ b: The edge and vertexdisjoint path problems together with their unsplittable ow generalization are NPha ..."
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Cited by 20 (2 self)
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In a packing integer program, we are given a matrix A and column vectors b; c with nonnegative entries. We seek a vector x of nonnegative integers, which maximizes c^T x; subject to Ax &le; b: The edge and vertexdisjoint path problems together with their unsplittable ow generalization are NPhard problems with a multitude of applications in areas such as routing, scheduling and bin packing. These two categories of problems are known to be conceptually related, but this connection has largely been ignored in terms of approximation algorithms. We explore the topic of approximating disjointpath problems using polynomialsize packing integer programs. Motivated by the...
A Shorter Proof of the Graph Minor Algorithm  The Unique Linkage Theorem
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful theorem which describes the structure of graphs excluding a fixed minor. This result is used to prove Wagner’s conjecture and provide a polynomial time algorithm for the disjoint paths problem when the number of the ..."
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Cited by 17 (6 self)
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful theorem which describes the structure of graphs excluding a fixed minor. This result is used to prove Wagner’s conjecture and provide a polynomial time algorithm for the disjoint paths problem when the number of the terminals is fixed (i.e, the Graph Minor Algorithm). However, both results require the full power of the Graph Minor Theory, i.e, the structure theorem. In this paper, we show that this is not true in the latter case. Namely, we provide a new and much simpler proof of the correctness of the Graph Minor Algorithm. Specifically, we prove the “Unique Linkage Theorem ” without using Graph Minors structure theorem. The new argument, in addition to being simpler, is much shorter, cutting the proof by at least 200 pages. We also give a new full proof of correctness of an algorithm for the wellknown edgedisjoint paths problem when the number of the terminals is fixed, which is at most 25 pages long.