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34
Simultaneous Optimization for Concave Costs: Single Sink Aggregation or Single Source BuyatBulk
 In Proc. of the 14 th Symposium on Discrete Algorithms (SODA
, 2003
"... We consider the problem of finding efficient trees to send information from k sources to a single sink in a network where information can be aggregated at intermediate nodes in the tree. Specifically, we assume that if information from j sources is traveling over a link, the total information tha ..."
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Cited by 101 (3 self)
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We consider the problem of finding efficient trees to send information from k sources to a single sink in a network where information can be aggregated at intermediate nodes in the tree. Specifically, we assume that if information from j sources is traveling over a link, the total information that needs to be transmitted is f(j). One natural and important (though not necessarily comprehensive) class of functions is those which are concave, nondecreasing, and satisfy f(0) = 0. Our goal is to find a tree which is a good approximation simultaneously to the optimum trees for all such functions. This problem is motivated by aggregation in sensor networks, as well as by buyatbulk network design.
Hardness of BuyatBulk Network Design
, 2004
"... We consider the BuyatBulk network design problem in which we wish to design a network for carrying multicommodity demands from a set of source nodes to a set of destination nodes. The key feature of the problem is that the cost of capacity on each edge is concave and hence exhibits economies of sc ..."
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Cited by 64 (4 self)
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We consider the BuyatBulk network design problem in which we wish to design a network for carrying multicommodity demands from a set of source nodes to a set of destination nodes. The key feature of the problem is that the cost of capacity on each edge is concave and hence exhibits economies of scale. If the cost of capacity per unit length can be different on different edges then we say that the problem is nonuniform. The problem is uniform otherwise.
Approximation algorithms for nonuniform buyatbulk network design problems
 Proc. of IEEE FOCS
"... Abstract. Buyatbulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorith ..."
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Cited by 58 (15 self)
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Abstract. Buyatbulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorithms for buyatbulk network design problems with costs on both edges and nodes of an undirected graph. Our main result is the first polylogarithmic approximation ratio for the nonuniform problem that allows different cost functions on each edge and node; the ratio we achieve is O(log4 h) where h is the number of demand pairs. In addition we present an O(log h) approximation for the single sink problem. Polylogarithmic ratios for some related problems are also obtained. Our algorithm for the multicommodity problem is obtained via a reduction to the single source problem using the notion of junction trees. We believe that this presents a simple yet useful general technique for network design problems. Key words. Nonuniform buyatbulk, network design, approximation algorithm, concave cost, network flow, economies of scale AMS subject classifications. 68Q25, 68W25, 90C27, 90C59 1. Introduction. Network
The PrizeCollecting Generalized Steiner Tree Problem Via A New Approach Of PrimalDual Schema
"... In this paper we study the prizecollecting version of the Generalized Steiner Tree problem. To the best of our knowledge, there is no general combinatorial technique in approximation algorithms developed to study the prizecollecting versions of various problems. These problems are studied on a cas ..."
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Cited by 46 (13 self)
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In this paper we study the prizecollecting version of the Generalized Steiner Tree problem. To the best of our knowledge, there is no general combinatorial technique in approximation algorithms developed to study the prizecollecting versions of various problems. These problems are studied on a case by case basis by Bienstock et al. [5] by applying an LProunding technique which is not a combinatorial approach. The main contribution of this paper is to introduce a general combinatorial approach towards solving these problems through novel primaldual schema (without any need to solve an LP). We fuse the primaldual schema with Farkas lemma to obtain a combinatorial 3approximation algorithm for the PrizeCollecting Generalized Steiner Tree problem. Our work also inspires a combinatorial algorithm [12] for solving a special case of Kelly’s problem [21] of pricing edges. We also consider the kforest problem, a generalization of kMST and kSteiner tree, and we show that in spite of these problems for which there are constant factor approximation algorithms, the kforest problem is much harder to approximate. In particular, obtaining an approximation factor better than O(n 1/6−ε) for kforest requires substantially new ideas including improving the approximation factor O(n 1/3−ε) for the notorious densest ksubgraph problem. We note that kforest and prizecollecting version of Generalized Steiner Tree are closely related to each other, since the latter is the Lagrangian relaxation of the former.
Bidimensionality: New Connections between FPT Algorithms and PTASs
"... We demonstrate a new connection between fixedparameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled “bidimensional” problems to show that essentially all such problems ha ..."
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Cited by 36 (5 self)
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We demonstrate a new connection between fixedparameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled “bidimensional” problems to show that essentially all such problems have both subexponential fixedparameter algorithms and PTASs. Bidimensional problems include e.g. feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertexremoval problems, dominating set, edge dominating set, rdominating set, diameter, connected dominating set, connected edge dominating set, and connected rdominating set. We obtain PTASs for all of these problems in planar graphs and certain generalizations; of particular interest are our results for the two wellknown problems of connected dominating set and general feedback vertex set for planar graphs and their generalizations, for which PTASs were not known to exist. Our techniques generalize and in some sense unify the two main previous approaches for designing PTASs in planar graphs, namely, the LiptonTarjan separator approach [FOCS’77] and the Baker layerwise decomposition approach [FOCS’83]. In particular, we replace the notion of separators with a more powerful tool from the bidimensionality theory, enabling the first approach to apply to a much broader class of minimization problems than previously possible; and through the use of a structural backbone and thickening of layers we demonstrate how the second approach can be applied to problems with a “nonlocal” structure.
On nonuniform multicommodity buyatbulk network design
 Proc. of ACM STOC
, 2005
"... We study the multicommodity buyatbulk network design problem in which we seek to design a network that satisfies the demands between terminals from a given set of sourcesink pairs. The key characteristic of this problem is the fact that the cost functions associated with the edges of the graph are ..."
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Cited by 35 (1 self)
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We study the multicommodity buyatbulk network design problem in which we seek to design a network that satisfies the demands between terminals from a given set of sourcesink pairs. The key characteristic of this problem is the fact that the cost functions associated with the edges of the graph are subadditive monotone and hence experience economies of scale. In the nonuniform case, each edge has its own cost function – possibly different from other edges. Special cases of this problem have been studied extensively: there are approximation algorithms when the edge cost functions are identical or when all sourcesink pairs share the same source. We present the first nontrivial approximation algorithm for the general case. Our algorithm is an extremely simple randomized greedy algorithm and has an approximation guarantee of exp(O ( √ ln nln ln n)) when the instance has at most n sourcesink pairs with unit demands. In the case of general demands, this yields an approximation factor of exp(O ( √ ln nln ln n)) log D, where D is the value of the largest demand.
Oblivious network design
 In Proceedings of the 17th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2006
"... Consider the following network design problem: given a network G = (V, E), sourcesink pairs {si, ti} arrive and desire to send a unit of flow between themselves. The cost of the routing is this: if edge e carries a total of fe flow (from all the terminal pairs), the cost is given by ∑ e ℓ(fe), wher ..."
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Cited by 31 (8 self)
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Consider the following network design problem: given a network G = (V, E), sourcesink pairs {si, ti} arrive and desire to send a unit of flow between themselves. The cost of the routing is this: if edge e carries a total of fe flow (from all the terminal pairs), the cost is given by ∑ e ℓ(fe), where ℓ is some concave cost function; the goal is to minimize the total cost incurred. However, we want the routing to be oblivious: when terminal pair {si, ti} makes its routing decisions, it does not know the current flow on the edges of the network, nor the identity of the other pairs in the system. Moreover, it does not even know the identity of the function ℓ, merely knowing that ℓ is a concave function of the total flow on the edge. How should it (obliviously) route its one unit of flow?
Simple cost sharing schemes for multicommodity rentorbuy and stochastic steiner tree
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... ..."
Approximating BuyatBulk and Shallowlight kSteiner trees
 In Proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems
, 2006
"... Abstract We study two related network design problems with two cost functions. In the buyatbulk kSteiner tree problem we are given a graph G(V, E) with a set of terminals T ` V including aparticular vertex s called the root, and an integer k < = T . There are two cost functions on theedges of G ..."
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Cited by 16 (2 self)
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Abstract We study two related network design problems with two cost functions. In the buyatbulk kSteiner tree problem we are given a graph G(V, E) with a set of terminals T ` V including aparticular vertex s called the root, and an integer k < = T . There are two cost functions on theedges of G, a buy cost b: E! R+ and a distance cost r: E! R+. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost Pe2H b(e)+Pt2Ts dist(t, s) is minimized, where dist(t, s) is the distance from t to s in H with respect to the r cost. We present an O(log4 n)approximation algorithm for the buyatbulk kSteiner tree problem. The second andclosely related one is bicriteria approximation algorithm for Shallowlight kSteiner trees. In theshallowlight kSteiner tree problem we are given a graph G with edge costs b(e) and distance costs r(e), and an integer k. Our goal is to find a minimum cost (under bcost) kSteiner tree such thatthe diameter under rcost is at most some given bound D. We develop an (O(log n), O(log3 n))approximation algorithm for a relaxed version of Shallowlight kSteiner tree where the solutionhas at least k 8 terminals. Using this we obtain an (O(log 2 n), O(log4 n))approximation algorithm