# Approximation Algorithms for the Maximum Internal Spanning Tree Problem

• Gábor Salamon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

## Abstract

We consider the MaximumInternalSpanningTree problem which is to find a spanning tree of a given graph with a maximum number of non-leaf nodes. From an optimization point of view, this problem is equivalent to the MinimumLeafSpanningTree problem, and is NP-hard as being a generalization of the HamiltonianPath problem. Although there is no constant factor approximation for the MinimumLeafSpanningTree problem [1], MaximumInternalSpanningTree can be approximated within a factor of 2 [2].

In this paper we improve this factor by giving a $$\frac{7}{4}$$-approximation algorithm. We also investigate the node-weighted case, when the weighted sum of the internal nodes is to be maximized. For this problem, we give a (2Δ− 3)-approximation for general graphs, and a 2-approximation for claw-free graphs. All our algorithms are based on local improvement steps.

## Keywords

Approximation algorithm Spanning tree leaves Hamiltonian path

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