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

Cristina Bazgan (1) , Sónia Toubaline (1) , Daniel Vanderpooten (1)

1 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision

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−ϵ , for any ϵ>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.

Mots clés: Complexity; Approximation; Most vital edges/nodes; Minimum spanning tree; Min edge/node blocker

Domaines: Informatique [cs]

Cristina Bazgan, Sónia Toubaline, Daniel Vanderpooten. Critical edges/nodes for the minimum spanning tree problem: complexity and approximation. Journal of Combinatorial Optimization, 2013, 26 (1), ⟨10.1007/s10878-011-9449-4⟩. ⟨hal-01499697⟩