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245
When trees collide: An approximation algorithm for the generalized Steiner problem on networks
, 1994
"... We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with linkcosts and, for each pair fi; jg of nodes, an edgeconnectivity requirement r ij . The goal is to find a minimumcost network using the a ..."
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Cited by 212 (31 self)
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We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with linkcosts and, for each pair fi; jg of nodes, an edgeconnectivity requirement r ij . The goal is to find a minimumcost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within 2dlog 2 (r + 1)e of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial minmax approximate equality relating minimumcost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.
Routing and Wavelength Assignment in AllOptical Networks
 IEEE/ACM Transactions on Networking
, 1995
"... This paper considers the problem of routing connections in a reconfigurable optical network using wavelength division multiplexing, where each connection between a pair of nodes in the network is assigned a path through the network and a wavelength on that path, such that connections whose paths sha ..."
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Cited by 207 (10 self)
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This paper considers the problem of routing connections in a reconfigurable optical network using wavelength division multiplexing, where each connection between a pair of nodes in the network is assigned a path through the network and a wavelength on that path, such that connections whose paths share a common link in the network are assigned different wavelengths. We derive an upper bound on the carried traffic of connections (or equivalently, a lower bound on the blocking probability) for any routing and wavelength assignment (RWA) algorithm in such a network. The bound scales with the number of wavelengths and is achieved asymptotically (when a large number of wavelengths is available) by a fixed RWA algorithm. Although computationally intensive, our bound can be used as a metric against which the performance of different RWA algorithms can be compared for networks of moderate size. We illustrate this by comparing the performance of a simple shortestpath RWA (SPRWA) algorithm via...
Greedy Randomized Adaptive Search Procedures For The Steiner Problem In Graphs
 QUADRATIC ASSIGNMENT AND RELATED PROBLEMS, VOLUME 16 OF DIMACS SERIES ON DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
, 1999
"... We describe four versions of a Greedy Randomized Adaptive Search Procedure (GRASP) for finding approximate solutions of general instances of the Steiner Problem in Graphs. Di#erent construction and local search algorithms are presented. Preliminary computational results with one of the versions ..."
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Cited by 104 (30 self)
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We describe four versions of a Greedy Randomized Adaptive Search Procedure (GRASP) for finding approximate solutions of general instances of the Steiner Problem in Graphs. Di#erent construction and local search algorithms are presented. Preliminary computational results with one of the versions on a variety of test problems are reported. On the majority of instances from the ORLibrary, a set of standard test problems, the GRASP produced optimal solutions. On those that optimal solutions were not found, the GRASP found good quality approximate solutions.
A new approach to the minimum cut problem
 Journal of the ACM
, 1996
"... Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds th ..."
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Cited by 98 (8 self)
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Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in O(n 2 log 3 n) time, a significant improvement over the previous Õ(mn) time bounds based on maximum flows. It is simple and intuitive and uses no complex data structures. Our algorithm can be parallelized to run in �� � with n 2 processors; this gives the first proof that the minimum cut problem can be solved in ���. The algorithm does more than find a single minimum cut; it finds all of them. With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of � of the minimum cut’s in expected Õ(n 2 � ) time, or in �� � with n 2 � processors. The problem of finding a minimum multiway cut of a graph into r pieces is solved in expected Õ(n 2(r�1) ) time, or in �� � with n 2(r�1) processors. The “trace ” of the algorithm’s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the
D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
 In: Surveys in Combinatorics, volume 327 of London Mathematical Society Lecture Note Series
, 2005
"... ..."
Global Mincuts in RNC, and Other Ramifications of a Simple MinCut Algorithm
, 1992
"... This paper presents a new algorithm for nding global mincuts in weighted, undirected graphs. One of the strengths of the algorithm is its extreme simplicity. This randomized algorithm can be implemented as a strongly polynomial sequential algorithm with running time ~ O(mn 2), even if space is res ..."
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Cited by 50 (5 self)
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This paper presents a new algorithm for nding global mincuts in weighted, undirected graphs. One of the strengths of the algorithm is its extreme simplicity. This randomized algorithm can be implemented as a strongly polynomial sequential algorithm with running time ~ O(mn 2), even if space is restricted to O(n), or can be parallelized as an RN C algorithm which runs in time O(log 2 n) on a CRCW PRAM with mn 2 log n processors. In addition to yielding the best known processor bounds on unweighted graphs, this algorithm provides the first proof that the mincut problem for weighted undirected graphs is in RN C. The algorithm does more than find a single mincut; it nds all of them. The algorithm also yields numerous results on network reliability, enumeration of cuts, multiway cuts, and approximate mincuts.
An NC Algorithm for Minimum Cuts
 IN PROCEEDINGS OF THE 25TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
"... We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from ..."
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Cited by 46 (3 self)
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We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from the minimum cut problem to the problem of obtaining a (2 + ffl)approximation to the minimum cut. This reduction involves a natural combinatorial SetIsolation Problem that can be solved easily in RNC. The third result is a derandomization of this RNC solution that requires a combination of two widely used tools: pairwise independence and random walks on expanders. We believe that the setisolation approach will prove useful in other derandomization problems. The techniques extend to two related problems: we describe NC algorithms finding minimum kway cuts for any constant k and finding all cuts of value within any constant factor of the minimum. Another application of these techni...
Predicting protein complex membership using probabilistic network reliability
 Genome Res
, 2004
"... data ..."
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the ..."
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Cited by 45 (11 self)
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I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
Path Set Selection in Mobile Ad Hoc Networks
, 2002
"... Topological changes in mobile ad hoc networks frequently render routing paths unusable. Such recurrent path failures have detrimental effects on the network ability to support QoSdriven services. A promising technique for addressing this problem is to use multiple redundant paths between the source ..."
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Cited by 44 (6 self)
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Topological changes in mobile ad hoc networks frequently render routing paths unusable. Such recurrent path failures have detrimental effects on the network ability to support QoSdriven services. A promising technique for addressing this problem is to use multiple redundant paths between the source and the destination. However,while multipath routing algorithms can tolerate network failures well,their failure resilience only holds if the paths are selected judiciously. In particular,the correlation between the failures of the paths in a redundant path set should be as small as possible. However,selecting an optimal path set is an NPcomplete problem. Heuristic solutions proposed in the literature are either too complex to be performed in realtime, or too ineffective,or both. This paper proposes a multipath routing algorithm,called Disjoint Pathset Selection Protocol (DPSP),based on a novel heuristic that,in nearly linear time on average,picks a set of highly reliable paths. The convergence to a highly reliable path set is very fast,and the protocol provides flexibility in path selection and routing algorithm. Furthermore,DPSP is suitable for realtime execution,with nearly no message exchange overhead and with minimal additional storage requirements. This paper presents evidence that multipath routing can mask a substantial number of failures in the network compared to single path routing protocols,and that the selection of paths according to DPSP can be beneficial for mobile ad hoc networks,since it dramatically reduces the rate of route discoveries.