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The complete optimal stars-clustering-tree problem

DISCRETE APPLIED MATHEMATICS 2008年第4期156卷 444-450页

作者： Korach, Ephraim Stern, Michal Ben Gurion Univ Negev IL-84105 Beer Sheva Israel Univ Haifa Caesarea Rothschild Inst IL-31999 Haifa Israel Acad Coll Tel Aviv Jaffa Tel Aviv Israel

We consider the following complete optimal stars-clustering-tree problem: Given a complete graph G = (V, E) with a weight on every edge and a collection of subsets of V, we want to find a minimum weight spanning tree T such that each subset of the vertices in the collection induces a complete star in T. One motivation for this problem is to construct a minimum cost (weight) communication tree network for a collection of (not necessarily disjoint) groups of customers such that each group induces a complete star. As a result the network will provide a "group broadcast" property, "group fault tolerance" and "group privacy". We present another motivation from database systems with replications. For the case where the intersection graph of the subsets is connected we present a structure theorem that describes all feasible solutions. Based on it we provide a polynomial algorithm for finding an optimal solution. For the case where each subset induces a complete star minus at most k leaves we prove that the problem is NP-hard. (C) 2007 Elsevier B.V. All rights reserved.

关键词： combinatorial optimization stars clustering spanning trees hypergraphs polynomial graph algorithms NP-hardness

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The clustering matroid and the optimal clustering tree

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MATHEMATICAL PROGRAMMING 2003年第1-3期98卷 385-414页

作者： Korach, E Stern, M Ben Gurion Univ Negev Dept Ind Engn & Management IL-84105 Beer Sheva Israel

We consider the following problem: Given a complete graph G=(V,E) with a cost on every edge and a given collection of subsets of V, we have to find a minimum cost spanning tree T such that each subset of the vertices in the collection induces a subtree in T. One motivation for this problem is to construct a minimum cost communication tree network for a collection of non-disjoint groups of customers such that the network will provide ``group fault tolerance'' and ``group privacy''. We model this problem as a matroid. We extend it to general matroids and call the new matroids ``clustering matroids''. We define three variations of the clustering tree problem and show that from an algorithmic point of view they are polynomially equivalent. We present a polynomial algorithm for one of the three variations, which implies that all of them can be solved polynomially. For the case where the cardinality of the subsets in the collection does not exceed three, we provide a greedy algorithm, a linear algorithm and also a polyhedron description of the convex hull of all the feasible solutions.

关键词： combinatorial optimization clustering spanning trees matroids hypergraphs polynomial graph algorithms primal-dual algorithms polyhedra

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The clustering matroid and the optimal clustering tree

The clustering matroid and the optimal clustering tree

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Workshop on Integer Programming

作者： Korach, E Stern, M Ben Gurion Univ Negev Dept Ind Engn & Management IL-84105 Beer Sheva Israel

关键词： combinatorial optimization clustering spanning trees matroids hypergraphs polynomial graph algorithms primal-dual algorithms polyhedra

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On the Optimal Stars Clustering Tree Problem

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Electronic Notes in Discrete Mathematics 2006年 27卷 99-100页

作者： Stern, Michal Korach, Ephraim Computer Science The Academic College of Tel Aviv-Yaffo Tel-Aviv Israel Industrial Engineering and Management Ben-Gurion University of the Negev Beer-Sheva Israel

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