The complete optimal stars-clustering-tree problem

Authors:
Ephraim Korach;Michal Stern
Affiliations:
Ben-Gurion University of the Negev, Beer-Sheva, Israel;Caesarea Rothschild Institute, University of Haifa, Haifa, Israel and The Academic College of Tel-Aviv-Jaffa, Israel
Venue:
Discrete Applied Mathematics
Year:
2008

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Abstract

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.