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50
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.
A new approach to the minimum cut problem
 Journal of the ACM
, 1996
"... Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds th ..."
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Cited by 126 (9 self)
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Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in O(n 2 log 3 n) time, a significant improvement over the previous Õ(mn) time bounds based on maximum flows. It is simple and intuitive and uses no complex data structures. Our algorithm can be parallelized to run in �� � with n 2 processors; this gives the first proof that the minimum cut problem can be solved in ���. The algorithm does more than find a single minimum cut; it finds all of them. With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of � of the minimum cut’s in expected Õ(n 2 � ) time, or in �� � with n 2 � processors. The problem of finding a minimum multiway cut of a graph into r pieces is solved in expected Õ(n 2(r�1) ) time, or in �� � with n 2(r�1) processors. The “trace ” of the algorithm’s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the
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
Global Mincuts in RNC, and Other Ramifications of a Simple MinCut Algorithm
, 1992
"... This paper presents a new algorithm for nding global mincuts in weighted, undirected graphs. One of the strengths of the algorithm is its extreme simplicity. This randomized algorithm can be implemented as a strongly polynomial sequential algorithm with running time ~ O(mn 2), even if space is res ..."
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Cited by 70 (5 self)
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This paper presents a new algorithm for nding global mincuts in weighted, undirected graphs. One of the strengths of the algorithm is its extreme simplicity. This randomized algorithm can be implemented as a strongly polynomial sequential algorithm with running time ~ O(mn 2), even if space is restricted to O(n), or can be parallelized as an RN C algorithm which runs in time O(log 2 n) on a CRCW PRAM with mn 2 log n processors. In addition to yielding the best known processor bounds on unweighted graphs, this algorithm provides the first proof that the mincut problem for weighted undirected graphs is in RN C. The algorithm does more than find a single mincut; it nds all of them. The algorithm also yields numerous results on network reliability, enumeration of cuts, multiway cuts, and approximate mincuts.
Tight lower bounds for certain parameterized NPhard problems
 Proc. 19th Annual IEEE Conference on Computational Complexity (CCC’04
, 2004
"... Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of wellknown NPhard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on deptht circuits cannot be solve ..."
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Cited by 65 (10 self)
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Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of wellknown NPhard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on deptht circuits cannot be solved in time no(k) poly(m), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t − 1)st level W [t − 1] of the Whierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NPhard problems, including weighted sat, dominating set, hitting set, set cover, and feature set, cannot be solved in time no(k) poly(m), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W [1] of the Whierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted qsat (for any fixed q ≥ 2), clique, and independent set, cannot be solved in time no(k) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n k poly(m) or O(n k). 1
Finding kcuts within Twice the Optimal
, 1995
"... Two simple approximation algorithms for the minimum kcut problem are presented. Each algorithm finds a kcut having weight within a factor of (2 \Gamma 2=k) of the optimal. One of our algorithms is particularly efficient  it requires a total of only n \Gamma 1 maximum flow computations for find ..."
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Cited by 48 (2 self)
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Two simple approximation algorithms for the minimum kcut problem are presented. Each algorithm finds a kcut having weight within a factor of (2 \Gamma 2=k) of the optimal. One of our algorithms is particularly efficient  it requires a total of only n \Gamma 1 maximum flow computations for finding a set of nearoptimal kcuts, one for each value of k between 2 and n. i 1 Introduction The minimum kcut problem is as follows: given an undirected graph G = (V; E) with nonnegative edge weights and a positive integer k, find a set S ` E of minimum weight whose removal leaves k connected components. This problem is of considerable practical significance, especially in the area of VLSI design. Solving this problem exactly is NPhard [GH], but no efficient approximation algorithms were known for it. In this paper we give two simple algorithms for finding kcuts. We prove a performance guarantee of (2 \Gamma 2=k) for each algorithm; however, neither algorithm dominates the other on a...
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...
Coclustering of image segments using convex optimization applied to EM neuronal reconstruction
 In CVPR
"... This paper addresses the problem of jointly clustering two segmentations of closely correlated images. We focus in particular on the application of reconstructing neuronal structures in oversegmented electron microscopy images. We formulate the problem of coclustering as a quadratic semiassignmen ..."
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Cited by 22 (2 self)
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This paper addresses the problem of jointly clustering two segmentations of closely correlated images. We focus in particular on the application of reconstructing neuronal structures in oversegmented electron microscopy images. We formulate the problem of coclustering as a quadratic semiassignment problem and investigate convex relaxations using semidefinite and linear programming. We further introduce a linear programming method with manageable number of constraints and present an approach for learning the cost function. Our method increases computational efficiency by orders of magnitude while maintaining accuracy, automatically finds the optimal number of clusters, and empirically tends to produce binary assignment solutions. We illustrate our approach in simulations and in experiments with real EM data. 1.