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106
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.
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 27 (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.
Optimizing the placement of integration points in multihop wireless networks
 IN PROCEEDINGS OF ICNP
, 2004
"... Efficient integration of a multihop wireless network with the Internet is an important research problem, and benefits several applications, such as wireless neighborhood networks and sensor networks. In a wireless neighborhood network, a few Internet Transit Access Points (ITAPs), serving as gatewa ..."
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Cited by 26 (2 self)
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Efficient integration of a multihop wireless network with the Internet is an important research problem, and benefits several applications, such as wireless neighborhood networks and sensor networks. In a wireless neighborhood network, a few Internet Transit Access Points (ITAPs), serving as gateways to the Internet, are deployed across the neighborhood; houses are equipped with lowcost antennas, and form a multihop wireless network among themselves to cooperatively route traffic to the Internet through the ITAPs. In a sensor network, sensors collect measurement data and send it through a multihop wireless network to the servers on the Internet via ITAPs. For both applications, placement of integration points between the wireless and wired network is a critical determinant of system performance and resource usage. However there has been little work on this subject. In this paper, we explore the placement problem under three wireless link models. For each link model, we develop algorithms to make informed placement decisions based on neighborhood layouts, user demands, and wireless link characteristics. We also extend our algorithms to provide fault tolerance and handle significant workload variation. We evaluate our placement algorithms using both analysis and simulation, and show that our algorithms yield close to optimal solutions over a wide range of scenarios we have considered.
On Network Design Problems: Fixed Cost Flows and the Covering Steiner Problem
, 2001
"... Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edgecostow problems. An edgecost ow problem is a mincost ow problem in which the cost of the ow equals the sum of the costs of the edges carrying positive ow. ..."
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Cited by 26 (3 self)
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Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edgecostow problems. An edgecost ow problem is a mincost ow problem in which the cost of the ow equals the sum of the costs of the edges carrying positive ow.
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 24 (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,
On the Hardness of 4coloring a 3colorable Graph
 In Proceedings of the 15th Annual IEEE Conference on Computational Complexity
, 2000
"... We give a new proof showing that it is NPhard to color a 3colorable graph using just four colors. This result is already known [19], but our proof is novel as it does not rely on the PCP theorem, while the one in [19] does. This highlights a qualitative difference between the known hardness res ..."
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Cited by 23 (2 self)
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We give a new proof showing that it is NPhard to color a 3colorable graph using just four colors. This result is already known [19], but our proof is novel as it does not rely on the PCP theorem, while the one in [19] does. This highlights a qualitative difference between the known hardness result for coloring 3colorable graphs and the factor n hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem [3].
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)).
TimeConstrained Scheduling of Weighted Packets on Trees and Meshes
 In Proceedings of 11th ACM Symposium on Parallel Algorithms and Architectures (SPAA
, 2003
"... The timeconstrained packet routing problem is to schedule a set of packets to be transmitted through a multinode network, where every packet has a source and a destination (as in traditional packet routing problems) as well as a release time and a deadline. The objective is to schedule the maximum ..."
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Cited by 20 (1 self)
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The timeconstrained packet routing problem is to schedule a set of packets to be transmitted through a multinode network, where every packet has a source and a destination (as in traditional packet routing problems) as well as a release time and a deadline. The objective is to schedule the maximum number of packets subject to deadline constraints. This problem is studied in [1], where it is shown that the problem is NPComplete even when the underlying topology is a linear array. Approximation algorithms are also provided in [1] for the linear array and the unidirectional ring for both the case where packets may be buffered in transit and the case where they may not be. In this paper we extend...
The Maximum EdgeDisjoint Paths Problem In Bidirected Trees
 SIAM Journal on Discrete Mathematics
, 1998
"... . A bidirected tree is the directed graph obtained from an undirected tree by replacing each undirected edge by two directed edges with opposite directions. Given a set of directed paths in a bidirected tree, the goal of the maximum edgedisjoint paths problem is to select a maximumcardinality subse ..."
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. A bidirected tree is the directed graph obtained from an undirected tree by replacing each undirected edge by two directed edges with opposite directions. Given a set of directed paths in a bidirected tree, the goal of the maximum edgedisjoint paths problem is to select a maximumcardinality subset of the paths such that the selected paths are edgedisjoint. This problem can be solved optimally in polynomial time for bidirected trees of constant degree, but is MAXSNPhard for bidirected trees of arbitrary degree. For every fixed " ? 0, a polynomialtime (5=3+ ")approximation algorithm is presented. Key words. approximation algorithms, edgedisjoint paths, bidirected trees AMS subject classifications. 68Q25, 68R10 1. Introduction. Research on disjoint paths problems in graphs has a long history [12]. In recent years, edgedisjoint paths problems have been brought into the focus of attention by advances in the field of communication networks. Many modern network architectures estab...