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Finding minimum 3way cuts in hypergraphs
"... Abstract. The minimum 3way cut problem in an edgeweighted hypergraph is to find a partition of the vertices into 3 sets minimizing the total weight of hyperedges with at least two endpoints in two different sets. In this paper we present some structural properties for minimum 3way cuts and design ..."
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Abstract. The minimum 3way cut problem in an edgeweighted hypergraph is to find a partition of the vertices into 3 sets minimizing the total weight of hyperedges with at least two endpoints in two different sets. In this paper we present some structural properties for minimum 3way cuts and design an O(dmn 3) algorithm for the minimum 3way cut problem in hypergraphs, where n and m are the numbers of vertices and edges respectively, and d is the sum of the degrees of all the vertices. Our algorithm is the first deterministic algorithm finding minimum 3way cuts in hypergraphs. 1
Complexity and Applications of EdgeInduced VertexCuts
, 2006
"... Motivated by hypergraph decomposition algorithms, we introduce the notion of edgeinduced vertexcuts and compare it with the wellknown notions of edgecuts and vertexcuts. We investigate the complexity of computing minimum edgeinduced vertexcuts and demonstrate the usefulness of our notion by ..."
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Motivated by hypergraph decomposition algorithms, we introduce the notion of edgeinduced vertexcuts and compare it with the wellknown notions of edgecuts and vertexcuts. We investigate the complexity of computing minimum edgeinduced vertexcuts and demonstrate the usefulness of our notion by applications in network reliability and constraint satisfaction.
The minimum Gc cut problem
"... Abstract. In this paper we study the complexity and approximability of the Gccut problem. Given a complete undirected graph Kn = (V; E) with V  = n, edge weighted by w(vi, vj) ≥ 0 and an undirected cluster graph, Gc = (Vc, Ec), with V c  = k, a kcut is a partition V1,..., Vk of V (G) such t ..."
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Abstract. In this paper we study the complexity and approximability of the Gccut problem. Given a complete undirected graph Kn = (V; E) with V  = n, edge weighted by w(vi, vj) ≥ 0 and an undirected cluster graph, Gc = (Vc, Ec), with V c  = k, a kcut is a partition V1,..., Vk of V (G) such that Vi = ∅ for i = 1,..., k. The Gccut problem is to compute a kcut minimizing P w(Vi, Vj) (i,j)∈Ec where w(Vi, Vj) = P p∈V w(p, q). Denote Gc as cluster graph and its vertices as clusters. We i,q∈Vj show that the Gccut problem is NPhard and even not approximable in the general case and remains NPhard for cluster trees. In particular, we give a complete characterization of hard cases for cluster graphs with at most four vertices by proving that the Gccut problem is NPhard if and only if Gc is isomorphic to 2K2. We also identify some cases where the Gccut problem is either polynomial or NPhard. Finally, we propose polynomial approximation results for the Gccut problem when the edge weights of G satisfy the triangle inequality, or when the weights are strictly positive.