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SIAM J. Discrete Math. 01/2006; 20:261-271.
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ABSTRACT: We consider the Steiner k-cut problem which generalizes both the k-cut problem and the multiway cut problem. The Steiner k-cut problem is defined as follows. Given an edge-weighted undirected graph G = (V, E), a subset of vertices X ⊆ V called terminals, and an integer k ≤ |X|, the objective is to find a minimum weight set of edges whose removal results in k dis-connected components, each of which contains at least one terminal. We give two approximation algorithms for the problem: a greedy (2 − 2 k)-approximation based on Gomory-Hu trees, and a (2− 2 |X|)-approximation based on LP rounding. We use the insight from the rounding to develop an exact bi-directed formulation for the global minimum cut problem (the k-cut problem with k = 2).
10/2005;
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ABSTRACT: We consider the Steiner k-cut problem, which is a common generalization of the k-cut problem and the multiway cut problem: given an edge-weighted undirected graph G = (V; E), a subset of vertices X ` V called terminals, and an integer k jXj, the objective is to find a minimum weight set of edges whose removal results in k disconnected components, each of which contains at least one terminal. We give two approximation algorithms for the problem: a 2 Gamma k -approximation based on Gomory-Hu trees, and a 2 Gamma jXj -approximation based on LP rounding.
05/2003;
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Automata, Languages and Programming, 30th International Colloquium, ICALP 2003, Eindhoven, The Netherlands, June 30 - July 4, 2003. Proceedings; 01/2003
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ABSTRACT: We consider the Steiner k-cut problem, which is a common generalization of the k-cut problem and the multiway cut problem: given an edge-weighted undirected graph G = (V,E), a subset of vertices X
$
\subseteq
$
\subseteq
V called terminals, and an integer k ≤ |X|, the objective is to find a minimum weight set of edges whose removal results in k disconnected components, each of which contains at least one terminal. We give two approximation algorithms for the problem:
a 2 − 2/k-approximation based on Gomory-Hu trees, and a 2 − 2/|X|-approximation based on LP rounding. The latter algorithm is based on rounding a generalization of a linear programming relaxation
suggested by Naor and Rabani [8]. The rounding uses the Goemans and Williamson primal-dual algorithm (and analysis) for the Steiner tree problem [4] in an interesting way and differs from the rounding in [8]. We use the insight from the rounding to develop an exact bi-directed formulation for the global minimum cut problem (the
k-cut problem with k = 2).
12/2002: pages 190-190;
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ABSTRACT: We give the first non-trivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work were the trivial O(k)-approximations. For the directed Steiner tree problem, we design a family of algorithms that achieves an approximation ratio of i(i Gamma 1)k 1=i in time O(n i k 2i ) for any fixed i ? 1, where k is the number of terminals. Thus, an O(k ffl ) approximation ratio can be achieved in polynomial time for any fixed ffl ? 0. Setting i = log k, we obtain an O(log 2 k) approximation ratio in quasi-polynomial time. For the directed generalized Steiner network problem, we give an algorithm that achieves an approximation ratio of O(k 2=3 log 1=3 k), where k is the number of pairs of vertices that are to be connected. Related problems including the group Steiner...
07/2000;
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ABSTRACT: We present improved combinatorial approximation algorithms for the uncapacitated facility location and k-median problems. Two central ideas in most of our results are cost scaling and greedy improvement. We present a simple greedy local search algorithm which achieves an approximation ratio of 2:414 + in ~ O(n 2 =) time. This also yields a bicriteria approximation tradeoff of (1 +; 1+ 2=) for facility cost versus service cost which is better than previously known tradeoffs and close to the best possible. Combining greedy improvement and cost scaling with a recent primal dual algorithm for facility location due to Jain and Vazirani, we get an approximation ratio of 1.853 in ~ O(n 3 ) time. This is already very close to the approximation guarantee of the best known algorithm which is LP-based. Further, combined with the best known LP-based algorithm for facility location, we get a very slight improvement in the approximation factor for facility location, achieving 1.728....
12/1999;
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ABSTRACT: Most optimization problems on an undirected graph reduce in complexity when restricted to instances on a tree. A recent result [3] for probabilistically approximating graph metrics by trees such that no edge stretches (in an expected sense) by more than a factor of O(log 2 n) has resulted in several approximation algorithms which exploit the ease of solving problems on trees. The tree construction in [3] is inherently randomized and a natural question to ask is whether approximation algorithms which use this construction can be derandomized. We present a general framework for derandomizing approximation algorithms which use the above tree construction as a primitive. Let Pi be a graph optimization problem which can be expressed as an integer program with 0-1 variables x(e) for each edge and with an objective function expressible as Supported by a Stanford Graduate Fellowship, an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger F...
11/1999;
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ABSTRACT: Most optimization problems on an undirected graph reduce in complexity when restricted to instances on a tree. A recent result [3] for probabilistically approximating graph metrics by trees such that no edge stretches (in an expected sense) by more than a factor of O(log 2 n) has resulted in several approximation algorithms which exploit the ease of solving problems on trees. The tree construction in [3] is inherently randomized and a natural question to ask is whether approximation algorithms which use this construction can be derandomized. We present a general framework for derandomizing approximation algorithms which use the above tree construction as a primitive. Let Pi be a graph optimization problem which can be expressed as an integer program with 0-1 variables x(e) for each edge and with an objective function expressible as Supported by a Stanford Graduate Fellowship, an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger F...
06/1999;
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Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, May 1-4, 1999, Atlanta, Georgia, USA; 01/1999
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[show abstract]
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ABSTRACT: Most optimization problems on an undirected graph reduce in complexity when restricted to instances on a tree. A recent result [3] for probabilistically approximating graph metrics by trees such that no edge stretches (in an expected sense) by more than a factor of O(log 2 n) has resulted in several approximation algorithms which exploit the ease of solving problems on trees. The tree construction in [3] is inherently randomized and a natural question to ask is whether approximation algorithms which use this construction can be derandomized. We present a general framework for derandomizing approximation algorithms which use the above tree construction as a primitive. Let Pi be a graph optimization problem which can be expressed as an integer program with 0-1 variables ¯ x(e) for each edge and with an objective function expressible as...
04/1998;
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Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998; 01/1998
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39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA; 01/1998
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ABSTRACT: We present the first constant-factor approximation algorithm for the metric k-median problem. The k-median problem is one of the most well-studied clustering problems, i.e., those problems in which the aim is to partition a given set of points into clusters so that the points within a cluster are relatively close with respect to some measure. For the metric k-median problem, we are given n points in a metric space. We select k of these to be cluster centers and then assign each point to its closest selected center. If point j is assigned to a center i, the cost incurred is proportional to the distance between i and j. The goal is to select the k centers that minimize the sum of the assignment costs. We give a -approximation algorithm for this problem. This improves upon the best previously known result of O(log k log log k), which was obtained by refining and derandomizing a randomized O(log n log log n)-approximation algorithm of Bartal.
Journal of Computer and System Sciences.
