Critical edges/nodes for the minimum spanning tree problem: complexity and approximation

Authors:
Cristina Bazgan;Sonia Toubaline;Daniel Vanderpooten
Affiliations:
LAMSADE, Université Paris-Dauphine, Paris Cedex 16, France 75775 and Institut Universitaire de France, Paris, France;LAMSADE, Université Paris-Dauphine, Paris Cedex 16, France 75775;LAMSADE, Université Paris-Dauphine, Paris Cedex 16, France 75775
Venue:
Journal of Combinatorial Optimization
Year:
2013

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Abstract

In this paper, we study the complexity and the approximation of the k most vital edges (nodes) and min edge (node) blocker versions for the minimum spanning tree problem (MST). We show that the k most vital edges MST problem is NP-hard even for complete graphs with weights 0 or 1 and 3-approximable for graphs with weights 0 or 1. We also prove that the k most vital nodes MST problem is not approximable within a factor n 1驴驴 , for any 驴0, unless NP=ZPP, even for complete graphs of order n with weights 0 or 1. Furthermore, we show that the min edge blocker MST problem is NP-hard even for complete graphs with weights 0 or 1 and that the min node blocker MST problem is NP-hard to approximate within a factor 1.36 even for graphs with weights 0 or 1.