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31
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 (10 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.
Secondnet: a data center network virtualization architecture with bandwidth guarantees
 In ACM CoNEXT
, 2010
"... In this paper, we propose virtual data center (VDC) as the unit of resource allocation for multiple tenants in the cloud. VDCs are more desirable than physical data centers because the resources allocated to VDCs can be rapidly adjusted as tenants ’ needs change. To enable the VDC abstraction, we de ..."
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Cited by 55 (3 self)
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In this paper, we propose virtual data center (VDC) as the unit of resource allocation for multiple tenants in the cloud. VDCs are more desirable than physical data centers because the resources allocated to VDCs can be rapidly adjusted as tenants ’ needs change. To enable the VDC abstraction, we designed a data center network virtualization architecture called SecondNet. SecondNet is scalable by distributing all the virtualtophysical mapping, routing, and bandwidth reservation state in server hypervisors. Its portswitching based source routing (PSSR) further makes SecondNet applicable to arbitrary network topologies using commodity servers and switches. SecondNet introduces a centralized VDC allocation algorithm for virtual to physical mapping with bandwidth guarantee. Simulations demonstrated that our VDC allocation achieves high network utilization and low time complexity. Our implementation and experiments on our testbed demonstrate that we can build SecondNet on top of various network topologies, and SecondNet provides bandwidth guarantee and elasticity, as designed. 1.
Rethinking Virtual Network Embedding: Substrate Support for Path Splitting and Migration
"... Network virtualization is a powerful way to run multiple architectures or experiments simultaneously on a shared infrastructure. However, making efficient use of the underlying resources requires effective techniques for virtual network embedding—mapping each virtual network to specific nodes and li ..."
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Cited by 54 (0 self)
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Network virtualization is a powerful way to run multiple architectures or experiments simultaneously on a shared infrastructure. However, making efficient use of the underlying resources requires effective techniques for virtual network embedding—mapping each virtual network to specific nodes and links in the substrate network. Since the general embedding problem is computationally intractable, past research restricted the problem space to allow efficient solutions, or focused on designing heuristic algorithms. In this paper, we advocate a different approach: rethinking the design of the substrate network to enable simpler embedding algorithms and more efficient use of resources, without restricting the problem space. In particular, we simplify virtual link embedding by: i) allowing the substrate network to split a virtual link over multiple substrate paths and ii) employing path migration to periodically reoptimize the utilization of the substrate network. We also explore nodemapping algorithms that are customized to common classes of virtualnetwork topologies. Our simulation experiments show that path splitting, path migration, and customized embedding algorithms enable a substrate network to satisfy a much larger mix of virtual networks.
Approximation algorithms for disjoint paths and related routing and packing problems
 Mathematics of Operations Research
, 2000
"... Abstract. Given a network and a set of connection requests on it, we consider the maximum edgedisjoint paths and related generalizations and routing problems that arise in assigning paths for these requests. We present improved approximation algorithms and/or integrality gaps for all problems consi ..."
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Cited by 54 (1 self)
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Abstract. Given a network and a set of connection requests on it, we consider the maximum edgedisjoint paths and related generalizations and routing problems that arise in assigning paths for these requests. We present improved approximation algorithms and/or integrality gaps for all problems considered; the central theme of this work is the underlying multicommodity flow relaxation. Applications of these techniques to approximating families of packing integer programs are also presented. Key words and phrases. Disjoint paths, approximation algorithms, unsplittable flow, routing, packing, integer programming, multicommodity flow, randomized algorithms, rounding, linear programming. 1
Fairness in routing and load balancing
 J. Comput. Syst. Sci
, 1999
"... We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria interacts in a complex fashion with the optimization of network utilization and throughput; in this work, we undertake an investigation of this relationship through the framework o ..."
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Cited by 53 (0 self)
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We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria interacts in a complex fashion with the optimization of network utilization and throughput; in this work, we undertake an investigation of this relationship through the framework of approximation algorithms. In a range of settings including both highspeed networks and Internet applications, maxmin fairness has emerged as a widely accepted formulation of the notion of fairness. Informally, we say that an allocation of bandwidth is maxmin fair if there is no way to give more bandwidth to any connection without decreasing the allocation to a connection of lesser or equal bandwidth. Given a collection of transmission routes, this criterion imposes a certain equilibrium condition on the bandwidth allocation, and some simple flow control mechanisms converge quickly to this equilibrium state. Indeed, the vast majority of previous work on maxmin fairness has focused on this issue of associating rates with connections that are specified by a fixed set of paths. Very little work has been devoted to understanding the relationship between the way in which one selects paths
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 50 (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 49 (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.
Computing Nash equilibria for scheduling on restricted parallel links
 In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC
, 2004
"... We consider the problem of routing n users on m parallel links, under the restriction that each user may only be routed on a link from a certain set of allowed links for the user. Thus, the problem is equivalent to the correspondingly restricted problem of assigning n jobs to m parallel machines. In ..."
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Cited by 48 (8 self)
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We consider the problem of routing n users on m parallel links, under the restriction that each user may only be routed on a link from a certain set of allowed links for the user. Thus, the problem is equivalent to the correspondingly restricted problem of assigning n jobs to m parallel machines. In a pure Nash equilibrium, no user may improve its own individual cost (delay) by unilaterally switching to another link from its set of allowed links. As our main result, we introduce a polynomial time algorithm to compute from any given assignment a pure Nash equilibrium with nonincreased makespan. The algorithm gradually changes a given assignment by pushing unsplittable user traffics through a network that is defined by the users and the links. Here, we use ideas from blocking flows. Furthermore, we use similar techniques as in the generic PreflowPush algorithm to approximate a schedule with minimum makespan, gaining an improved approximation factor of 2 − 1 for identical links, where w1 is the largest user traffic. w1 We extend this result to related links, gaining an approximation factor of 2. Our approximation algorithms run in polynomial time. We close with tight upper bounds on the coordination ratio for pure Nash equilibria.
The AllorNothing Multicommodity Flow Problem
 in Proceedings of the 36th ACM Symposium on Theory of Computing (STOC
, 2004
"... m)), 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 40 (10 self)
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m)), 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. \Lambda
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 39 (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