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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 194 (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.
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 71 (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
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 45 (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...
A Faster Algorithm for Computing Minimum 5Way and 6Way Cuts in Graphs
 In 5th Annual International Computing and Combinatorics Conference
, 1999
"... For an edgeweighted graph G with n vertices and m edges, the minimum kway cut problem is to find a partition of the vertex set into k nonempty subsets so that the weight sum of edges between different subsets is minimized. For this problem with k = 5 and 6, we present a deterministic algorithm th ..."
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Cited by 7 (1 self)
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For an edgeweighted graph G with n vertices and m edges, the minimum kway cut problem is to find a partition of the vertex set into k nonempty subsets so that the weight sum of edges between different subsets is minimized. For this problem with k = 5 and 6, we present a deterministic algorithm that runs in k02 (nF (n; m)+C 2 (n; m)+n )) = O(mn =m)) time, where F(n, m) and C_2(n, m) denote respectively the time bounds required to solve the maximum flow problem and the minimum 2way cut problem in G. The bounds ~ ) for k = 5 and ) for k = 6 improve the previous best randomized bounds ), respectively.
Fast Randomized Algorithms for Computing Minimum {3,4,5,6}Way Cuts
"... A minimum kway cut of an nvertex, medge, weighted, undirected graph is a partition of the vertices into k sets that minimizes the total weight of edges with endpoints in dierent sets. We give new randomized algorithms to nd minimum 3way and 4way cuts, which lead to time bounds of O(mn k 2 log ..."
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Cited by 4 (0 self)
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A minimum kway cut of an nvertex, medge, weighted, undirected graph is a partition of the vertices into k sets that minimizes the total weight of edges with endpoints in dierent sets. We give new randomized algorithms to nd minimum 3way and 4way cuts, which lead to time bounds of O(mn k 2 log 3 n) time for k 6. This improves on the best previous time bounds by a factor of ~ n 2 ). 1 Introduction A minimum kway cut of an nvertex, medge, weighted, undirected graph is a partition of the vertices into k sets that minimizes the total weight of edges with endpoints in dierent sets. Asking for a minimum kway cut is equivalent to asking for the edge set of minimum total weight whose removal would break the graph into at least k connected components. Our main motivations for studying minimum kway cuts are that they are a natural property of graphs, and that they have received considerable attention in the past. Nagamochi and Ibaraki [11] also point to a number of appl...