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59
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 108 (12 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.
Hardness of the undirected edgedisjoint paths problem
 Proc. of STOC
, 2005
"... In the EdgeDisjoint Paths problem with Congestion (EDPwC), we are given a graph with n nodes, a set of terminal pairs and an integer c. The objective is to route as many terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge in the graph. When c ..."
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Cited by 56 (8 self)
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In the EdgeDisjoint Paths problem with Congestion (EDPwC), we are given a graph with n nodes, a set of terminal pairs and an integer c. The objective is to route as many terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge in the graph. When c = 1, the problem is simply referred to as the EdgeDisjoint Paths (EDP) problem. In this paper, we study the hardness of EDPwC in undirected graphs. We obtain an improved hardness result for EDP, and also show the first polylogarithmic integrality gaps and hardness of approximation results for EDPwC. Specifically, we prove that EDP is (log 1 2 −ε n)hard to approximate for any constant ε> 0, unless NP ⊆ ZP T IME(n polylog n). We also show that for any congestion c = o(log log n / log log log n), there is no (log 1−ε c+1 n)approximation algorithm for EDPwC, unless NP ⊆ ZP T IME(npolylog n). For larger congestion, where c ≤ η log log n / log log log n for some constant η, we obtain superconstant inapproximability ratios. All of our hardness results can be converted into integrality gaps for the multicommodity flow relaxation. We also present a separate elementary direct proof of this integrality gap result. Finally, we note that similar results can be obtained for the AllorNothing Flow (ANF) problem, a relaxation of EDP, in which the flow unit routed between the sourcesink pairs does not have follow a single path, so the resulting flow is not necessarily integral. Using standard transformations, our results also extend to the nodedisjoint versions of these problems as well as to the directed setting. 1
Improved Bounds for the Unsplittable Flow Problem
 In Proceedings of the 13th ACMSIAM Symposium on Discrete Algorithms
, 2002
"... In this paper we consider the unsplittable ow problem (UFP): given a directed or undirected network G = (V, E) with edge capacities and a set of terminal pairs (or requests) with associated demands, find a subset of the pairs of maximum total demand for which a single flow path can be chosen for eac ..."
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Cited by 56 (6 self)
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In this paper we consider the unsplittable ow problem (UFP): given a directed or undirected network G = (V, E) with edge capacities and a set of terminal pairs (or requests) with associated demands, find a subset of the pairs of maximum total demand for which a single flow path can be chosen for each pair so that for every edge, the sum of the demands of the paths crossing the edge does not exceed its capacity.
Approximation Algorithms for the Unsplittable Flow Problem
"... We present approximation algorithms for the unsplittable flow problem (UFP) on undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the nonuniform capacity case in which the edge capacities can vary arbitrarily ..."
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Cited by 55 (9 self)
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We present approximation algorithms for the unsplittable flow problem (UFP) on undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the nonuniform capacity case in which the edge capacities can vary arbitrarily over the graph. Our results are: For undirected graphs we obtain a O(\Delta ff \Gamma 1 log2 n) approximation ratio, where n is the number of vertices, \Delta the maximum degree, and ff the expansion of the graph. Our ratio is capacity independent and improves upon the earlier O(\Delta ff \Gamma 1(c max=cmin) log n) bound [15] for large values of cmax=cmin. Furthermore, if we specialize to the case where all edges have the same capacity, our algorithm gives an O(\Delta ff \Gamma 1 log n) approximation, which matches the performance of the bestknown algorithm [15] for this special case. For certain strong constantdegree expanders considered by Frieze [10] we obtain an O(plog n) approximation for the uniform capacity case, improving upon the current O(log n) approximation. For UFP on the line and the ring, we give the first constantfactor approximation algorithms. Previous results addressed only the uniform capacity case. All of the above results improve if the maximum demand is bounded
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.
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.
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,
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.
Models and Techniques for Communication in Dynamic Networks Christian
 In Proc. of the 19th Symp. on Theoretical Aspects of Computer Science (STACS
, 2001
"... In this paper we will present various models and techniques for communication in dynamic networks. Dynamic networks are networks of dynamically changing bandwidth or topology. Situations in which dynamic networks occur are, for example: faulty networks (links go up and down), the Internet (the bandw ..."
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Cited by 23 (2 self)
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In this paper we will present various models and techniques for communication in dynamic networks. Dynamic networks are networks of dynamically changing bandwidth or topology. Situations in which dynamic networks occur are, for example: faulty networks (links go up and down), the Internet (the bandwidth of connections may vary), and wireless networks (mobile units move around). We investigate the problem of how to ensure connectivity, how to route, and how to perform admission control in these networks. Some of these problems have already been partly solved, but many problems are still wide open. The aim of this paper is to give an overview of recent results in this area, to identify some of the most interesting open problems and to suggest models and techniques that allow us to study them.
New hardness results congestion minimization and machine scheduling
 PROC. 36TH. ANNUAL ACM SYMPOSIUM ON THEORY OF COMPTING
"... We study the approximability of two natural NPhard problems. The first problem is congestion minimization in directed networks. In this problem, we are given a directed graph and a set of sourcesink pairs. The goal is to route all the pairs with minimum congestion on the network edges. The second ..."
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Cited by 23 (3 self)
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We study the approximability of two natural NPhard problems. The first problem is congestion minimization in directed networks. In this problem, we are given a directed graph and a set of sourcesink pairs. The goal is to route all the pairs with minimum congestion on the network edges. The second problem is machine scheduling, where we are given a set of jobs, and for each job, there is a list of intervals on which it can be scheduled. The goal is to find the smallest number of machines on which all jobs can be scheduled such that no two jobs overlap in their execution on any machine. Both problems are known to be O(log n/loglog n)approximable via the randomized rounding technique of Raghavan and Thompson. However, until recently, only Max SNP hardness was known for each problem. We make progress in closing this gap by showing that both problems are Ω(log log n)hard to approximate unless NP ⊆ DTIME(n O(log log log n)).