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An algorithmic framework for the exact solution of the prizecollecting Steiner tree problem
 MATHEMATICAL PROGAMMING, SERIES B
, 2006
"... The PrizeCollecting Steiner Tree Problem (PCST) on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. PCST appears frequently in the design of utility ne ..."
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Cited by 43 (14 self)
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The PrizeCollecting Steiner Tree Problem (PCST) on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. PCST appears frequently in the design of utility networks where profit generating customers and the network connecting them have to be chosen in the most profitable way. Our main contribution is the formulation and implementation of a branchandcut algorithm based on a directed graph model where we combine several stateoftheart methods previously used for the Steiner tree problem. Our method outperforms the previously published results on the standard benchmark set of problems. We can solve all benchmark instances from the literature to optimality, including some of them for which the optimum was not known. Compared to a recent algorithm by Lucena and Resende, our new method is faster by more than two orders of magnitude. We also introduce a new class of more challenging instances and present computational results for them. Finally, for a set of largescale realworld instances arising in the design of fiber optic networks, we also obtain optimal solution values.
Reductions Among High Dimensional Proximity Problems
, 2000
"... We present improved running times for a wide range of approximate high dimensional proximity problems. We obtain subquadratic running time for each of these problems. These improved running times are obtained by reduction to Nearest Neighbour queries. The problems we consider in this paper are Ap ..."
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Cited by 36 (4 self)
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We present improved running times for a wide range of approximate high dimensional proximity problems. We obtain subquadratic running time for each of these problems. These improved running times are obtained by reduction to Nearest Neighbour queries. The problems we consider in this paper are Approximate Diameter, Approximate Furthest Neighbours, Approximate Discrete Center, Approximate Line Center, Approximate Metric Facility Location, Approximate Bottleneck Matching, and Approximate Minimum Weight Matching. University of Southern California. Email: agoel@cs.usc.edu . y Stanford University. Email: indyk@cs.stanford.edu . z University of Iowa. Email: kvaradar@cs.uiowa.edu . 0 Problem Ref Approx. Time Comments Diameter [10] p 3 O(dn) [12] 1 + ffl O(dn log n + n 2 ) [2] 1 + ffl ~ O(n 2\GammaO(ffl 2 ) + dn) [18] 1 + ffl ~ O(n 1+1=(1+ffl=6) + dn) here 1 + ffl ~ O(n 1+1=(1+ffl) + dn) ~ O(n) (1 + ffl)NNS queries here p 2 ~ O(dn) see Section 3 for some e...
On the equivalence between the primaldual schema and the local ratio technique
 In 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX). Number 2129 in Lecture Notes in Computer Science
, 2001
"... Abstract. We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primaldual schema and the local ratio technique. Recently, primaldual algorithms were devised by first constructing a local ratio algorithm and then transform ..."
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Cited by 31 (8 self)
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Abstract. We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primaldual schema and the local ratio technique. Recently, primaldual algorithms were devised by first constructing a local ratio algorithm and then transforming it into a primaldual algorithm. This was done in the case of the 2approximation algorithms for the feedback vertex set problem and in the case of the first primaldual algorithms for maximization problems. Subsequently, the nature of the connection between the two paradigms was posed as an open question by Williamson [Math. Program., 91 (2002), pp. 447–478]. In this paper we answer this question by showing that the two paradigms are equivalent.
On the Approximability of Some Network Design Problems
, 2005
"... Consider the following classical network design problem: aset of terminals T = ftig wants to send traffic to a "root" r in an nnode graph G = (V; E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocatedon the edges to permit this. However, bandwidth on an ..."
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Cited by 30 (3 self)
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Consider the following classical network design problem: aset of terminals T = ftig wants to send traffic to a "root" r in an nnode graph G = (V; E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocatedon the edges to permit this. However, bandwidth on an edge e can only be allocated in integral multiples of some basecapacity ue and hence provisioning k \Theta ue bandwidth onedge e incurs a cost of dke times the cost of that edge. Theobjective is a minimumcost feasible solution. This is one of many network design problems widelystudied where the bandwidth allocation being governed by side constraints: edges may only allow a subset of cables tobe purchased on them, or certain qualityofservice requirements may have to be met.In this work, we show that the above problem, and in fact, several basic problems in this general network designframework, cannot be approximated better than \Omega (log log n)unless NP ` DTIME \Gamma nO(log log log n) \Delta. In particular,
Efficient Algorithms Using The Multiplicative Weights Update Method
, 2006
"... Abstract Algorithms based on convex optimization, especially linear and semidefinite programming, are ubiquitous in Computer Science. While there are polynomial time algorithms known to solve such problems, quite often the running time of these algorithms is very high. Designing simpler and more eff ..."
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Cited by 28 (1 self)
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Abstract Algorithms based on convex optimization, especially linear and semidefinite programming, are ubiquitous in Computer Science. While there are polynomial time algorithms known to solve such problems, quite often the running time of these algorithms is very high. Designing simpler and more efficient algorithms is important for practical impact. In this thesis, we explore applications of the Multiplicative Weights method in the design of efficient algorithms for various optimization problems. This method, which was repeatedly discovered in quite diverse fields, is an algorithmic technique which maintains a distribution on a certain set of interest, and updates it iteratively by multiplying the probability mass of elements by suitably chosen factors based on feedback obtained by running another algorithm on the distribution. We present a single metaalgorithm which unifies all known applications of this method in a common framework. Next, we generalize the method to the setting of symmetric matrices rather than real numbers. We derive the following applications of the resulting Matrix Multiplicative Weights algorithm: 1. The first truly general, combinatorial, primaldual method for designing efficient algorithms for semidefinite programming. Using these techniques, we obtain significantly faster algorithms for obtaining O(plog n) approximations to various graph partitioning problems, such as Sparsest Cut, Balanced Separator in both directed and undirected weighted graphs, and constraint satisfaction problems such as Min UnCut and Min 2CNF Deletion. 2. An ~O(n3) time derandomization of the AlonRoichman construction of expanders using Cayley graphs. The algorithm yields a set of O(log n) elements which generates an expanding Cayley graph in any group of n elements. 3. An ~O(n3) time deterministic O(log n) approximation algorithm for the quantum hypergraph covering problem. 4. An alternative proof of a result of Aaronson that the flfatshattering dimension of quantum states on n qubits is O ( nfl2).
Strategic Network Formation through Peering and Service Agreements
, 2010
"... We introduce a game theoretic model of network formation in an effort to understand the complex system of business relationships between various Internet entities (e.g., Autonomous Systems, enterprise networks, residential customers). This system is at the heart of Internet connectivity. In our mode ..."
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Cited by 26 (5 self)
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We introduce a game theoretic model of network formation in an effort to understand the complex system of business relationships between various Internet entities (e.g., Autonomous Systems, enterprise networks, residential customers). This system is at the heart of Internet connectivity. In our model we are given a network topology of nodes and links where the nodes act as the players of the game, and links represent potential contracts. Nodes wish to satisfy their demands, which earn potential revenues, but may have to pay their neighbors for links incident to them. We incorporate some of the qualities of Internet business relationships, including customerprovider and peering contracts. We show that every Nash equilibrium can be represented by a circulation flow of utility with certain constraints. This allows us to prove that the price of stability is at most 2 with respect to a natural objective function, but that prices of anarchy and stability can both be unbounded with respect to social welfare. We thus focus on the quality of equilibria achievable through centralized incentives, and show that if every payout is increased by a factor of 2, then there is a Nash equilibrium as good as the original centrally defined social optimum.
Approximation via Cost Sharing: Simpler and better approximation algorithms for network design
, 2005
"... We present constantfactor approximation algorithms for several widelystudied NPhard optimization problems in network design, including the multicommodity rentorbuy, virtual private network design, and singlesink buyatbulk problems. Our algorithms are simple and their approximation ratios imp ..."
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Cited by 26 (3 self)
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We present constantfactor approximation algorithms for several widelystudied NPhard optimization problems in network design, including the multicommodity rentorbuy, virtual private network design, and singlesink buyatbulk problems. Our algorithms are simple and their approximation ratios improve over those previously known, in some cases by orders of magnitude. We develop a general analysis framework to bound the approximation ratios of our algorithms. This framework is based on a novel connection between random sampling and gametheoretic cost sharing. While techniques from approximation algorithms have recently yielded new progress on costsharing problems, our work is the first to show the conversethat ideas from cost sharing can be fruitfully applied in the design and analysis of approximation algorithms.
Approximating Steiner Networks with Node Weights
, 2007
"... The (undirected) Steiner Network problem is: given a graph G = (V, E) with edge/node weights and connectivity requirements {r(u, v) : u, v ∈ U ⊆ V}, find a minimum weight subgraph H of G containing U so that the uvedgeconnectivity in H is at least r(u, v) for all u, v ∈ U. The seminal paper of Jai ..."
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Cited by 26 (12 self)
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The (undirected) Steiner Network problem is: given a graph G = (V, E) with edge/node weights and connectivity requirements {r(u, v) : u, v ∈ U ⊆ V}, find a minimum weight subgraph H of G containing U so that the uvedgeconnectivity in H is at least r(u, v) for all u, v ∈ U. The seminal paper of Jain [19], and numerous papers preceding it, considered the EdgeWeighted Steiner Network problem, with weights on the edges only, and developed novel tools for approximating minimum weight edgecovers of several types of set functions and families. However, for the NodeWeighted Steiner Network (NWSN) problem, nontrivial approximation algorithms were known only for 0, 1 requirements. We make an attempt to change this situation, by giving the first nontrivial approximation algorithm for NWSN with arbitrary requirements. Our approximation ratio for NWSN is rmax · O(ln U), where rmax = maxu,v∈U r(u, v). This generalizes the result of Klein and Ravi [21] for the case rmax = 1. We also give an O(ln U)approximation algorithm for the nodeconnectivity variant of NWSN (when the paths are required to be internallydisjoint) for the case rmax = 2. Our results are based on a much more general approximation algorithm for the problem of finding a minimum nodeweighted edgecover of an uncrossable setfamily. We also give the first evidence that a polylogarithmic approximation ratio for NWSN might not exist even for U  = 2 and unit weights. 1 1
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 25 (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.