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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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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.
Finding k cuts within twice the optimal
 SIAM Journal on Computing
, 1995
"... Abstract. Two simple approximation algorithms for the minimum kcut problem are presented. Each algorithm finds a k cut having weight within a factor of (2 2/k) of the optimal. One algorithm is particularly efficientit requires a total of only n maximum flow computations for finding a set of near ..."
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Abstract. Two simple approximation algorithms for the minimum kcut problem are presented. Each algorithm finds a k cut having weight within a factor of (2 2/k) of the optimal. One algorithm is particularly efficientit requires a total of only n maximum flow computations for finding a set of nearoptimal k cuts, one for each value of k between 2 and n. Key words, graph partitioning, minimum cuts, approximation algorithms AMS subject classifications. 68Q20, 68Q25
Mimicking networks and succinct representations of terminal cuts
 CoRR
"... Given a large edgeweighted network G with k terminal vertices, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of G is to construct a mimi ..."
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Cited by 3 (2 self)
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Given a large edgeweighted network G with k terminal vertices, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of G is to construct a mimicking network: a small network G ′ with the same k terminals, in which the minimum cut value between every bipartition of terminals is the same as in G. This notion was introduced by Hagerup, Katajainen, Nishimura, and Ragde [JCSS ’98], who proved that such G ′ of size at most 22k always exists. Obviously, by having access to the smaller network G ′, certain computations involving cuts can be carried out much more efficiently. We provide several new bounds, which together narrow the previously known gap from doublyexponential to only singlyexponential, both for planar and for general graphs. Our first and main result is that every kterminal planar network admits a mimicking network G ′ of size O(k 2 2 2k), which is moreover a minor of G. On the other hand, some planar networks G require E(G ′)  ≥ Ω(k 2). For general networks, we show that certain bipartite graphs only admit mimicking networks of size V (G ′)  ≥ 2 Ω(k) , and moreover, every data structure that stores the minimum cut value between all bipartitions of the terminals must use 2Ω(k) machine words. 1
is intuition.
"... I would like to express my deep appreciation and my sincere gratitude to my Masters thesis advisor, Prof. Robert Krauthgamer, for his kindly and patient guidance, friendly support and for his devoted attention. It was a pleasure to work and to be inspired by him. I would like to thank my fellow stud ..."
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I would like to express my deep appreciation and my sincere gratitude to my Masters thesis advisor, Prof. Robert Krauthgamer, for his kindly and patient guidance, friendly support and for his devoted attention. It was a pleasure to work and to be inspired by him. I would like to thank my fellow students, the faculty members and the administrative staff of the Department of Computer Science and Applied Mathematics for the pleasant atmosphere and the convenient workspace. Given a large edgeweighted network G with k vertices designated as terminals, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of G is to construct a mimicking network: a small network G ′ with the same k terminals, in which the minimum cut value between every bipartition of terminals is the same as in G. This notion was introduced by Hagerup, Katajainen, Nishimura, and Ragde [JCSS ’98], who proved that such G ′ of size
Partitioning a graph in alliances and its . . .
, 2004
"... Any reasonably large group of individuals, families, states, and parties exhibits the phenomenon of subgroup formations within the group such that the members of each group have a strong connection or bonding between each other. The reasons of the formation of these subgroups that we call alliances ..."
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Any reasonably large group of individuals, families, states, and parties exhibits the phenomenon of subgroup formations within the group such that the members of each group have a strong connection or bonding between each other. The reasons of the formation of these subgroups that we call alliances differ in different situations, such as, kinship and friendship (in the case of individuals), common economic interests (for both individuals and states), common political interests, and geographical proximity. This structure of alliances is not only prevalent in social networks, but it is also an important characteristic of similarity networks of natural and unnatural objects. (A similarity network defines the links between two objects based on their similarities). Discovery of such structure in a data set is called clustering or unsupervised learning and the ability to do it automatically is desirable for many applications in the areas of pattern recognition, computer vision, artificial intelligence, behavioral and social sciences, life sciences, earth sciences, medicine, and information theory. In this dissertation, we study a graph theoretical model of alliances where an alliance of the vertices of a graph is a set of vertices in the graph, such that every vertex in the set