D. Karger and M. Minkoff, Building Steiner trees with incomplete global knowledge, Proceedings of 41st IEEE FOCS, 2000.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to either problem above does not use the fact that the edge weights form a metric. Thus one may hope for a better approximation ratio using a di erent approach. Indeed our problem is a special case of the connected facility location problem. This problem is rst studied by Karger and Minko [10]. It generalizes our problem by adding a constant scaling factor M to the costs of all edges in the solution that connect two non leaf nodes . Later Gupta et al. improved the approximation ratio to 12 [8] We show that the General Steiner Tree Star problem (where the sets X and Y need not be ....

D. R. Karger and M. Minko. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, pages 613-623, (2000).

.... gives an evengreater improvement over previous combinatorial approximation algorithms for the problem [12] Related Work The connected facility location problem has received considerable recent attention both in the operations research literature [17, 19] and in the computer science community [13, 15, 16]. In addition to modeling the basic scenario of facility location in which some infrastructure among facilities must also be built, the problem naturally arises as a subroutine in several network design algorithms (see [13, 15] Karger and Minkoff [15] motivated by the so called maybecast ....

.... research literature [17, 19] and in the computer science community [13, 15, 16] In addition to modeling the basic scenario of facility location in which some infrastructure among facilities must also be built, the problem naturally arises as a subroutine in several network design algorithms (see [13, 15]) Karger and Minkoff [15] motivated by the so called maybecast problem, gave the first constant factor approximation algorithm for the problem. This algorithm is simple and combinatorial, but has a relatively large performance guarantee. Gupta et al. 13] subsequently employed an LP rounding ....

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David R. Karger and Maria Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41th Annual IEEE Symposium on Foundations of Computer Science, pages 613--623, 2000.

....is to optimize the cost of the facilities we open, and the total distance we have to ship the demand. This problem arises naturally in application such as the placement of warehouses [6] and caches on the web [13, 2, 15] It also arises as a subroutine in solving several network design problems [8, 11, 9]. We consider the variant of this problem where each facility has hard upper and lower bounds on the amount of demand it can serve. This is a natural assumption in many situations. For example, if a facility is a supermarket in a chain, we may not want overcrowding of any particular store. On the ....

....programming based approach. Though our approach needs to blow up the capacities by a constant factor, it has the advantage of showing a constant integrality gap. We feel this will have applications in cases where this algorithm is required as a subroutine in some bigger network design problem [2, 15, 9, 8, 11]. In the analysis of algorithms for these problems, it is often simple to construct fractional solutions to the sub problems from the optimal solution, and therefore, a good integrality gap is essential. Our Techniques: Non uniform capacities make the application of standard rounding techniques ....

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D. Karger and M. Minkoff. Building steiner trees with incomplete global knowledge. Proceedings of 41st IEEE FOCS, 2000.

....problem with decreasing cost functions can be helpful. The load balanced (a.k.a. lower bounded) facility location problem in an example. This problem, which is a special case of UniFL with f i (u) u 0] u l i ] for given f i and l i , was first defined by Karger and Minkoff [11] and Guha et al. 9] and used to solve other location problems. We still do not know of any constant factor approximation algorithm for this problem, and more generally for UniFL with decreasing cost functions. As in the hard capacitated facility location problem, the integrality gap of the ....

D. Karger and M. Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, 2000.

....cost functions is of special interest. However, other types of cost functions also occur in several applications. For example, many variants of the facility location problem such as soft or hard capacitated facility location [5, 6, 15, 22] and lower bounded (a.k.a. load balanced) facility location [10, 14] are special cases of this. This paper is organized as follows. In Section 2, we give a formal de nition of the problem, and observe that it can be reduced to both the capacitated and the uncapacitated facility location problems. In Section 3, we present a greedy 1.861 approximation algorithm ....

D. R. Karger and M. Minko, Building Steiner trees with incomplete global knowledge, In Proceedings of the 41st IEEE Symposium on Foundations of Computer Science 2000, 613-623.

.... on capacitated facility location, where either the capacities are hard [6, 13, 19] or where we are allowed multiple copies of a facility at a location [5, 10] Facility location variants arise in numerous situations purchasing cables on a network [8] constructing probabilistic Steiner trees [11], constructing multicommodity ow networks with incomplete knowledge [9] placing objects in caches with replication [2, 17] min sum clustering [3] to name a few. Our Techniques: Our algorithms are based on formulating the various problems as integer programs, and rounding the linear ....

D. Karger and M. Minko. Building steiner trees with incomplete global knowledge. Proceedings of 41st IEEE FOCS, 2000.

....we mention only the primal dual 3 approximation algorithm of Jain and Vazirani [20] and the currently best, 1. 52 approximation algorithm of Mahdian et al. 26] The first constant factor approximation algorithm for the Single Source Rent or Buy Network Design problem is due to Karger and Minkoff [23] and Guha, Meyerson and Munagala [16] see also [17] Both algorithms use a variant of facility location to gather the clients, and then build a Steiner tree on the set of gathering points. Swamy and Kumar [32] gave a primal dual 4.55 approximation algorithm, that is also built analogously as a ....

....we can find in polynomial time. It is not hard to see that the bought edges in the optimum rent or buy network must form a Steiner tree with s at its root, and that the rental edges form a shortest path connection from each user J to the closest point in the tree. Several approximation algorithms [23, 16, 32] for this problem exploit the optimal structure to construct a solution in two phases: first they use a facility location like algorithm to gather the users in a few center locations, and rent a path from each user to such a center. When enough demand is gathered at each center, a Steiner tree is ....

D. Karger and M. Minkoff. Building Steiner trees with incomplete global knowledge. Proceedings of the 41st Annual Symposium on Foundations of Computer Science, 2000.

....in layers. We will illustrate the construction for layer i. Let S i be the set of demand points we have at this stage. S1 is the original set of demand points. We include s in all the sets S i . Layer i will use pipe type i exclusively. We will use the load balanced facility location problem [9, 10] as a sub routine below. This problem is a variant of the classical facility location problem, where we have a lower bound on the amount of demand any open facility must serve. We can approximate this to a constant factor of r provided we relax the lower bound by factor = 1 1 . Here, r is ....

D. Karger and M. Minko. Building steiner trees with incomplete global knowledge. Proceedings of 41st IEEE FOCS, 2000.

....facilities in increasing order of their r, and close those which have less than r fi 2 fi assigned points. The points that were assigned there are sent to their respective closest open facilities. The proof of the following lemma is similar to the proof for load balanced facility location in [4, 2]. Lemma 3.1 Closing facilities does not increase the total cost of the solution. Proof: Suppose we close a facility i (r) because it did not have enough points assigned to it. Let D be the distance in the modified metric to the closest point j which was not connected to i (r) Then, j was ....

D. Karger and M. Minkoff. Building steiner trees with incomplete global knowledge. IEEE FOCS, 2000. 2

....for assigning demand node j to facility i. The new feature is that we incur an additional cost: we must also choose a Steiner tree T which connects all the open facilities, and we pay an additional cost proportional to the total edge length of T . This problem was introduced by Karger and Minkoff [15], who developed an algorithm approximating the optimum to within a constant factor. We provide a significantly improved constantfactor approximation algorithm for the Connected Facility Location problem, adapting a rounding technique of Shmoys, Tardos, and Aardal [22] Combined with our underlying ....

....x e can be at most the capacity u e . In this case we show that even checking feasibility of SymT and SymG is hard. We also give a bicriteria approximation algorithm for SymF in this case. Connections to Related Work. The connected facility location problem was first studied by Karger and Minkoff [15], who obtained a constant factor approximation algorithm. Their approach works in two stages. The first stage decides which facilities to open by clustering demands as much as possible (see also the paper by Guha et al. 12] In the second stage, they prove that it does not cost much to ....

David R. Karger and Maria Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41th Annual IEEE Symposium on Foundations of Computer Science, pages 613--623, 2000.

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D. Karger and M. Minkoff, Building Steiner trees with incomplete global knowledge, Proceedings of 41st IEEE FOCS, 2000.

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David R. Karger and Maria Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41th Annual IEEE Symposium on Foundations of Computer Science, pages 613--623, 2000.

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David R. Karger and Maria Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41th Annual IEEE Symposium on Foundations of Computer Science, pages 613--623, 2000. 11

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D. Karger and M. Minkoff. Building Steiner trees with incomplete global knowledge. 41st IEEE FOCS, 2000.

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D. Karger and M. Minko. Building steiner trees with incomplete global knowledge. IEEE Symposium on Foundations of Computer Science, 2000.

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D. R. Karger and M. Minko. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 613-623, 2000.

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D. Karger and M. Minko. Building steiner trees with incomplete global knowledge. Proceedings of 41st IEEE Symposium on Foundations of Computer Science, 2000.

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D. R. Karger and M. Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of 41st FOCS, pages 613--623, 2000.

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D. R. Karger and M. Minkoff. Building Steiner trees with incomplete global knowledge. In 41st FOCS, pages 613--623, 2000.

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David R. Karger and Maria Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41th Annual IEEE Symposium on Foundations of Computer Science, pages 613--623, 2000.

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David R. Karger and Maria Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41th Annual IEEE Symposium on Foundations of Computer Science, pages 613-- 623, 2000.

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D. R. Karger and M. Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of 41st FOCS, pages 613--623, 2000.

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D. R. Karger and M. Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, pages 613-623, (2000).

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D. Karger and M. Minkoff. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, 2000.

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David Karger and Maria Minko. Building steiner trees with incomplete global knowledge. Manuscript, 1999.

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