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Optimal 3Terminal Cuts and Linear Programming
"... . Given an undirected graph G = (V; E) and three specified terminal nodes t1 ; t2 ; t3 , a 3cut is a subset A of E such that no two terminals are in the same component of GnA. If a nonnegative edge weight ce is specified for each e 2 E, the optimal 3cut problem is to find a 3cut of minimum total ..."
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. Given an undirected graph G = (V; E) and three specified terminal nodes t1 ; t2 ; t3 , a 3cut is a subset A of E such that no two terminals are in the same component of GnA. If a nonnegative edge weight ce is specified for each e 2 E, the optimal 3cut problem is to find a 3cut of minimum total weight. This problem is NPhard, and in fact, is maxSNPhard. An approximation algorithm having performance guarantee 7 6 has recently been given by Calinescu, Karloff, and Rabani. It is based on a certain linear programming relaxation, for which it is shown that the optimal 3cut has weight at most 7 6 times the optimal LP value. It is proved here that 7 6 can be improved to 12 11 , and that this is best possible. As a consequence, we obtain an approximation algorithm for the optimal 3cut problem having performance guarantee 12 11 . 1 Introduction Given an undirected graph G = (V; E) and k specified terminal nodes t 1 ; : : : ; t k , a kcut is a subset A of E such that no two term...
Greedy splitting algorithms for approximating multiway partition problems
 Math. Programming
, 2005
"... Abstract. Given a system (V, T, f, k), where V is a finite set, T ⊆ V, f: 2 V → R is a submodular function and k ≥ 2 is an integer, the general multiway partition problem (MPP) asks to find a kpartition P = {V1, V2,..., Vk} of V that satisfies Vi ∩T � = ∅ for all i and minimizes f(V1)+f(V2)+ · · ..."
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Abstract. Given a system (V, T, f, k), where V is a finite set, T ⊆ V, f: 2 V → R is a submodular function and k ≥ 2 is an integer, the general multiway partition problem (MPP) asks to find a kpartition P = {V1, V2,..., Vk} of V that satisfies Vi ∩T � = ∅ for all i and minimizes f(V1)+f(V2)+ · · ·+f(Vk), where P is a kpartition of V if (i) Vi � = ∅, (ii) Vi ∩ Vj = ∅, i � = j, and (iii) V1 ∪ V2 ∪ · · · ∪ Vk = V hold. MPP formulation captures a generalization in submodular systems of many NPhard problems such as kway cut, multiterminal cut, target split and their generalizations in hypergraphs. This paper presents a simple and unified framework for developing and analyzing approximation algorithms for various MPPs. Key words. approximation algorithm – hypergraph partition – kway cut – multiterminal cut – multiway partition problem – submodular function 1.
An analysis of graph cut size for transductive learning
 In In proceedings of the 23rd International Conference on Machine Learning (ICML
, 2006
"... I consider the setting of transductive learning of vertex labels in graphs, in which a graph with n vertices is sampled according to some unknown distribution; there is a true labeling of the vertices such that each vertex is assigned to exactly one of k classes, but the labels of only some (r ..."
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I consider the setting of transductive learning of vertex labels in graphs, in which a graph with n vertices is sampled according to some unknown distribution; there is a true labeling of the vertices such that each vertex is assigned to exactly one of k classes, but the labels of only some (random) subset of the vertices are revealed to the learner.
Cut problems in graphs with a budget constraint
 IN PROC. 7TH LATIN AMERICAN THEORETICAL INFORMATICS SYMPOSIUM
, 2006
"... We study budgeted variants of classical cut problems: the Multiway Cut problem, the Multicut problem, and the kCut problem, and provide approximation algorithms for these problems. Specifically, for the budgeted multiway cut and the kcut problems we provide constant factor approximation algorithms ..."
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We study budgeted variants of classical cut problems: the Multiway Cut problem, the Multicut problem, and the kCut problem, and provide approximation algorithms for these problems. Specifically, for the budgeted multiway cut and the kcut problems we provide constant factor approximation algorithms. We show that the budgeted multicut problem is at least as hard to approximate as the sparsest cut problem, and we provide a bicriteria approximation algorithm for it.
Approximation algorithms for submodular multiway partition
 CoRR
"... Abstract — We study algorithms for the SUBMODULAR MUL ..."
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Abstract — We study algorithms for the SUBMODULAR MUL
Approximation algorithms for requirement cut on graphs
 In APPROX + RANDOM
, 2005
"... In this paper, we unify several graph partitioning problems including multicut, multiway cut, and kcut, into a single problem. The input to the requirement cut problem is an undirected edgeweighted graph G = (V, E), and g groups of vertices X1, · · · , Xg ⊆ V, with each group Xi having a requir ..."
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In this paper, we unify several graph partitioning problems including multicut, multiway cut, and kcut, into a single problem. The input to the requirement cut problem is an undirected edgeweighted graph G = (V, E), and g groups of vertices X1, · · · , Xg ⊆ V, with each group Xi having a requirement ri between 0 and Xi. The goal is to find a minimum cost set of edges whose removal separates each group Xi into at least ri disconnected components. We give an O(log n · log(gR)) approximation algorithm for the requirement cut problem, where n is the total number of vertices, g is the number of groups, and R is the maximum requirement. We also show that the integrality gap of a natural LP relaxation for this problem is bounded by O(log n · log(gR)). On trees, we obtain an improved guarantee of O(log(gR)). There is an Ω(log g) hardness of approximation for the requirement cut problem, even on trees. 1
Approximation and Hardness Results for Label Cut and Related Problems
"... We investigate a natural combinatorial optimization problem called the Label Cut problem. Given an input graph G with a source s and a sink t, the edges of G are classified into different categories, represented by a set of labels. The labels may also have weights. We want to pick a subset of labels ..."
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We investigate a natural combinatorial optimization problem called the Label Cut problem. Given an input graph G with a source s and a sink t, the edges of G are classified into different categories, represented by a set of labels. The labels may also have weights. We want to pick a subset of labels of minimum cardinality (or minimum total weight), such that the removal of all edges with these labels disconnects s and t. We give the first nontrivial approximation and hardness results for the Label Cut problem. Firstly, we present an O ( √ m)approximation algorithm for the Label Cut problem, where m is the number of edges in the input graph. Secondly, we show that it is NPhard to approximate Label Cut within 2 log1−1 / log logc n n for any constant c < 1/2, where n is the input length of the problem. Thirdly, our techniques can be applied to other previously considered optimization problems. In particular we show that the Minimum Label Path problem has the same approximation hardness as that of Label Cut, simultaneously improving and unifying two known hardness results for this problem which were previously the best (but incomparable due to different complexity assumptions). 1
Mathematical Morphology and Graphs: Application to Interactive Medical Image Segmentation
, 2008
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Web Scale Taxonomy Cleansing
"... Large ontologies and taxonomies are automatically harvested from webscale data. These taxonomies tend to be huge, noisy, and contains little context. As a result, cleansing and enriching those largescale taxonomies becomes a great challenge. A natural way to enrich a taxonomy is to map the taxonomy ..."
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Large ontologies and taxonomies are automatically harvested from webscale data. These taxonomies tend to be huge, noisy, and contains little context. As a result, cleansing and enriching those largescale taxonomies becomes a great challenge. A natural way to enrich a taxonomy is to map the taxonomy to existing datasets that contain rich information. In this paper, we study the problem of matching two web scale taxonomies. Besides the scale of the problem, we address the challenge that the taxonomies may not contain enough context (such as attribute values). As existing entity resolution techniques are based directly or indirectly on attribute values as context, we must explore external evidence for entity resolution. Specifically, we explore positive and negative evidence in external data sources such as the web and in other taxonomies. To integrate positive and negative evidence, we formulate the entity resolution problem as a problem of finding optimal multiway cuts in a graph. We analyze the complexity of the problem, and propose a Monte Carlo algorithm for finding greedy cuts. We conduct extensive experiments and compare our approach with three existing methods to demonstrate the advantage of our approach. 1.