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48
Polynomial Time Approximation Schemes for Dense Instances of NPHard Problems
, 1995
"... We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NPhard optimization problems, including maximum cut, graph bisection, graph separation, minimum kway cut with and without specified terminals, and maximum 3satisfiabi ..."
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Cited by 195 (32 self)
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We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NPhard optimization problems, including maximum cut, graph bisection, graph separation, minimum kway cut with and without specified terminals, and maximum 3satisfiability. By dense graphs we mean graphs with minimum degree &Omega;(n), although our algorithms solve most of these problems so long as the average degree is &Omega;(n). Denseness for nongraph problems is defined similarly. The unified framework begins with the idea of exhaustive sampling: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs where the objective function and the constraints are "dense" polynomials of constant degree.
The Complexity of Multiterminal Cuts
 SIAM Journal on Computing
, 1994
"... In the Multiterminal Cut problem we are given an edgeweighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, maxflow problem, and ..."
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Cited by 190 (0 self)
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In the Multiterminal Cut problem we are given an edgeweighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, maxflow problem, and can be solved in polynomial time. We show that the problem becomes NPhard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NPhard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2  2/k of the optimal cut weight.
Approximating the minimumdegree Steiner tree to within one of optimal
 JOURNAL OF ALGORITHMS
, 1994
"... ... some optimal tree for the respective problems. Unless P = N P, this is the best bound achievable in polynomial time. ..."
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Cited by 88 (6 self)
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... some optimal tree for the respective problems. Unless P = N P, this is the best bound achievable in polynomial time.
An improved approximation algorithm for multiway cut
 Journal of Computer and System Sciences
, 1998
"... Given an undirected graph with edge costs and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. Multiway Cut is the problem of finding a multiway cut of minimum cost. Previously, a very simple combinatorial algorithm due ..."
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Cited by 74 (5 self)
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Given an undirected graph with edge costs and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. Multiway Cut is the problem of finding a multiway cut of minimum cost. Previously, a very simple combinatorial algorithm due to Dahlhaus, � Johnson, Papadimitriou, Seymour, and Yannakakis gave a performance guarantee of 2 1 − 1 k. In this paper, we present a new linear programming relaxation for Multiway Cut and a new approximation algorithm based on it. The algorithm breaks the threshold of 2 for approximating Multiway Cut, achieving a. This improves the previous result for every value of k. performance ratio of at most 1.5 − 1 k In particular, for k = 3 we get a ratio of 7
Graph clustering and minimum cut trees
 Internet Mathematics
, 2004
"... Abstract. In this paper, we introduce simple graph clustering methods based on minimum cuts within the graph. The clustering methods are general enough to apply to any kind of graph but are well suited for graphs where the link structure implies a notion of reference, similarity, or endorsement, suc ..."
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Cited by 69 (4 self)
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Abstract. In this paper, we introduce simple graph clustering methods based on minimum cuts within the graph. The clustering methods are general enough to apply to any kind of graph but are well suited for graphs where the link structure implies a notion of reference, similarity, or endorsement, such as web and citation graphs. We show that the quality of the produced clusters is bounded by strong minimum cut and expansion criteria. We also develop a framework for hierarchical clustering and present applications to realworld data. We conclude that the clustering algorithms satisfy strong theoretical criteria and perform well in practice. 1.
Balanced graph partitioning
 In 16th Annual ACM Symposium on Parallelism in Algorithms and Architectures
, 2004
"... We consider the problem of partitioning a graph into k components of roughly equal size while minimizing the capacity of the edges between different components of the cut. In particular we require that for a parameter ν ≥ 1, no component contains more than ν · n k of the graph vertices. For k = 2 an ..."
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Cited by 67 (0 self)
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We consider the problem of partitioning a graph into k components of roughly equal size while minimizing the capacity of the edges between different components of the cut. In particular we require that for a parameter ν ≥ 1, no component contains more than ν · n k of the graph vertices. For k = 2 and ν = 1 this problem is equivalent to the well known Minimum Bisection Problem for which an approximation algorithm with a polylogarithmic approximation guarantee has been presented in [FK02]. For arbitrary k and ν ≥ 2 a bicriteria approximation ratio of O(logn) was obtained by [ENRS99] using the spreading metrics technique. We present a bicriteria approximation algorithm that for any constant ν> 1 runs in polynomial time and guarantees an approximation ratio of O(log1.5 n) (for a precise statement of the main result see Theorem 6). For ν = 1 and k ≥ 3 we show that no polynomial time approximation algorithm can guarantee a finite approximation ratio unless P = NP. 1
Multiway Cuts in Directed and Node Weighted Graphs
 in Proc. 21st ICALP, Lecture Notes in Computer Science 820
, 1994
"... this paper we consider node multiway cuts; the problem of computing a minimum weight node multiway cut is known to be NPhard and max SNPhard [1]. It turns out that the approximation algorithm in [2] for edge multiway cuts does not extend to the node multiway cut problem. Let us give a reason for t ..."
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Cited by 48 (4 self)
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this paper we consider node multiway cuts; the problem of computing a minimum weight node multiway cut is known to be NPhard and max SNPhard [1]. It turns out that the approximation algorithm in [2] for edge multiway cuts does not extend to the node multiway cut problem. Let us give a reason for this. Define an isolating cut for terminal s i to be a cut that separates s i from the rest of the terminals. A minimum isolating cut for s i can be computed in polynomial time by identifying the remaining terminals, and finding a minimum cut separating them from s i . The algorithm in [2] finds such cuts for each terminal, discards the heaviest cut, and picks the union of the remaining. The approximation factor is proven by observing that on doubling each edge in the optimum multiway cut, we can partition these edges into k isolating cuts, one for each Department of Computer Science and Engg., Indian Institute of Technology, New Delhi, India
Clustering query refinements by user intent
 In 19th International World Wide Web Conference, WWW
, 2010
"... We address the problem of clustering the refinements of a user search query. The clusters computed by our proposed algorithm can be used to improve the selection and placement of the query suggestions proposed by a search engine, and can also serve to summarize the different aspects of information r ..."
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Cited by 38 (0 self)
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We address the problem of clustering the refinements of a user search query. The clusters computed by our proposed algorithm can be used to improve the selection and placement of the query suggestions proposed by a search engine, and can also serve to summarize the different aspects of information relevant to the original user query. Our algorithm clusters refinements based on their likely underlying user intents by combining document click and session cooccurrence information. At its core, our algorithm operates by performing multiple random walks on a Markov graph that approximates user search behavior. A user study performed on top search engine queries shows that our clusters are rated better than corresponding clusters computed using approaches that use only document click or only sessions cooccurrence information. 1.
Increasing the Weight of Minimum Spanning Trees
, 1996
"... The problems of computing the maximum increase in the weight of the minimum spanning trees of a graph caused by the removal of a given number of edges, or by finite increases in the weights of the edges, are investigated. For the case of edge removals, the problem is shown to be NPhard and an \Omeg ..."
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Cited by 28 (1 self)
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The problems of computing the maximum increase in the weight of the minimum spanning trees of a graph caused by the removal of a given number of edges, or by finite increases in the weights of the edges, are investigated. For the case of edge removals, the problem is shown to be NPhard and an \Omega\Gamma/ = log k)approximation algorithm is presented for it, where k is the number of edges to be removed. The second problem is studied assuming that the increase in the weight of an edge has an associated cost proportional to the magnitude of the change. An O(n 3 m 2 log(n 2 =m)) time algorithm is presented to solve it. 1 Introduction Consider a communication network in which information is broadcast over a minimum spanning tree. There are applications for which it is important to determine the maximum degradation in the performance of the broadcasting protocol that can be expected as a result of traffic fluctuations and link failures [25]. Also, there are several combinatorial op...