Results 1  10
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21
The primaldual method for approximation algorithms and its application to network design problems.
, 1997
"... Abstract In this survey, we give an overview of a technique used to design and analyze algorithms that provide approximate solutions to N P hard problems in combinatorial optimization. Because of parallels with the primaldual method commonly used in combinatorial optimization, we call it the prim ..."
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Cited by 137 (5 self)
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Abstract In this survey, we give an overview of a technique used to design and analyze algorithms that provide approximate solutions to N P hard problems in combinatorial optimization. Because of parallels with the primaldual method commonly used in combinatorial optimization, we call it the primaldual method for approximation algorithms. We show how this technique can be used to derive approximation algorithms for a number of different problems, including network design problems, feedback vertex set problems, and facility location problems.
Clustering with qualitative information
 In Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
, 2003
"... We consider the problem of clustering a collection of elements based on pairwise judgments of similarity and dissimilarity. Bansal, Blum and Chawla [1] cast the problem thus: given a graph G whose edges are labeled “+ ” (similar) or “− ” (dissimilar), partition the vertices into clusters so that ..."
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Cited by 122 (9 self)
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We consider the problem of clustering a collection of elements based on pairwise judgments of similarity and dissimilarity. Bansal, Blum and Chawla [1] cast the problem thus: given a graph G whose edges are labeled “+ ” (similar) or “− ” (dissimilar), partition the vertices into clusters so that the number of pairs correctly (resp. incorrectly) classified with respect to the input labeling is maximized (resp. minimized). Complete graphs, where the classifier labels every edge, and general graphs, where some edges are not labeled, are both worth studying. We answer several questions left open in [1] and provide a sound overview of clustering with qualitative information. We give a factor 4 approximation for minimization on complete graphs, and a factor O(log n) approximation for general graphs. For the maximization version, a PTAS for complete graphs is shown in [1]; we give a factor 0.7664 approximation for general graphs, noting that a PTAS is unlikely by proving APXhardness. We also prove the APXhardness of minimization on complete graphs. 1.
Feedback set problems
 HANDBOOK OF COMBINATORIAL OPTIMIZATION
, 1999
"... ABSTRACT. This paper is a short survey of feedback set problems. It will be published in ..."
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Cited by 56 (1 self)
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ABSTRACT. This paper is a short survey of feedback set problems. It will be published in
On the minimum feedback vertex set problem: Exact and enumeration algorithms
, 2007
"... We present a time O(1.7548 n) algorithm finding a minimum feedback vertex set in an undirected graph on n vertices. We also prove that a graph on n vertices can contain at most 1.8638 n minimal feedback vertex sets and that there exist graphs having 105 n/10 ≈ 1.5926 n minimal feedback vertex sets. ..."
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Cited by 25 (14 self)
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We present a time O(1.7548 n) algorithm finding a minimum feedback vertex set in an undirected graph on n vertices. We also prove that a graph on n vertices can contain at most 1.8638 n minimal feedback vertex sets and that there exist graphs having 105 n/10 ≈ 1.5926 n minimal feedback vertex sets.
Multicuts in Unweighted Graphs and Digraphs with Bounded Degree and Bounded TreeWidth
, 1998
"... this paper. Also, we show that Directed Edge Multicut is NPhard in digraphs with treewidth one and maximum in and out degree three. Other hardness results indicate why we cannot eliminate any of the three restrictionsunweighted, bounded degree and bounded treewidthon the input graph and sti ..."
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Cited by 24 (0 self)
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this paper. Also, we show that Directed Edge Multicut is NPhard in digraphs with treewidth one and maximum in and out degree three. Other hardness results indicate why we cannot eliminate any of the three restrictionsunweighted, bounded degree and bounded treewidthon the input graph and still obtain a PTAS. It is known [1] that for a Max SNPhard problem, unless P=NP, no PTAS exists. We have already seen that Unweighted Edge Multicut is Max SNPhard in stars [9], so letting the input graph have unbounded degree makes the problem harder. We show that Weighted Edge Multicut is Max SNPhard in binary trees, therefore letting the input graph be weighted makes the problem harder. Finally, we show that Unweighted Edge Multicut is Max SNPhard if the input graphs are walls. Walls, to be formally defined in Section 6, have degree at most three and unbounded treewidth. We conclude that letting the input graph have unbounded treewidth makes the problem significantly harder
PrimalDual Approximation Algorithms for Feedback Problems in Planar Graphs
 IPCO '96
, 1996
"... Given a subset of cycles of a graph, we consider the problem of finding a minimumweight set of vertices that meets all cycles in the subset. This problem generalizes a number of problems, including the minimumweight feedback vertex set problem in both directed and undirected graphs, the subset fee ..."
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Cited by 23 (3 self)
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Given a subset of cycles of a graph, we consider the problem of finding a minimumweight set of vertices that meets all cycles in the subset. This problem generalizes a number of problems, including the minimumweight feedback vertex set problem in both directed and undirected graphs, the subset feedback vertex set problem, and the graph bipartization problem, in which one must remove a minimumweight set of vertices so that the remaining graph is bipartite. We give a 9/4approximation algorithm for the general problem in planar graphs, given that the subset of cycles obeys certain properties. This results in 9/4approximation algorithms for the aforementioned feedback and bipartization problems in planar graphs. Our algorithms use the primaldual method for approximation algorithms as given in Goemans and Williamson [16]. We also show that our results have an interesting bearing on a conjecture of Akiyama and Watanabe [2] on the cardinality of feedback vertex sets in planar graphs.
Multiway cuts in node weighted graphs
 JOURNAL OF ALGORITHMS
, 2004
"... A (2 — 2/k) approximation algorithm is presented for the node multiway cut problem, thus matching the result of Dahlhaus et al. (SIAM J. Comput. 23 (4) (1994) 864894) for the edge version of this problem. This is done by showing that the associated LPrelaxation always has a halfintegral optimal s ..."
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Cited by 20 (0 self)
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A (2 — 2/k) approximation algorithm is presented for the node multiway cut problem, thus matching the result of Dahlhaus et al. (SIAM J. Comput. 23 (4) (1994) 864894) for the edge version of this problem. This is done by showing that the associated LPrelaxation always has a halfintegral optimal solution.
Hitting diamonds and growing cacti
 in Proceedings of the 14th Conference on Integer Programming and Combinatorial Optimization (IPCO 2010
, 2010
"... Abstract. We consider the following NPhard problem: in a weighted graph, find a minimum cost set of vertices whose removal leaves a graph in which no two cycles share an edge. We obtain a constantfactor approximation algorithm, based on the primaldual method. Moreover, we show that the integralit ..."
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Cited by 7 (2 self)
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Abstract. We consider the following NPhard problem: in a weighted graph, find a minimum cost set of vertices whose removal leaves a graph in which no two cycles share an edge. We obtain a constantfactor approximation algorithm, based on the primaldual method. Moreover, we show that the integrality gap of the natural LP relaxation of the problem is Θ(logn), where n denotes the number of vertices in the graph.