Approximation Algorithms for the Maximum Internal Spanning Tree Problem

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


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.


Approximation algorithm Spanning tree leaves Hamiltonian path 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Lu, H.I., Ravi, R.: The power of local optimization: Approximation algorithms for maximum-leaf spanning tree (DRAFT). Technical Report CS-96-05, Department of Computer Science, Brown University, Providence, Rhode Island (1996)Google Scholar
  2. 2.
    Salamon, G., Wiener, G.: On finding spanning trees with few leaves (submitted 2006)Google Scholar
  3. 3.
    Solis-Oba, R.: 2-approximation algorithm for finding a spanning tree with maximum number of leaves. In: Bilardi, G., Pietracaprina, A., Italiano, G.F., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 441–452. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  4. 4.
    Schrijver, A.: 50: Shortest spanning trees. In: Combinatorial optimization. In: Polyhedra and efficiency, vol. B, pp. 855–876. Springer, Heidelberg (2003)Google Scholar
  5. 5.
    Zhang, S., Wang, Z.: Scattering number in graphs. Networks 37, 102–106 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Salamon, G., Wiener, G.: Leaves of spanning trees and vulnerability. In: The 5th Hungarian-Japanese Symposium on Discrete Mathematics and Its Applications, pp. 225–235 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Gábor Salamon
    • 1
  1. 1.Department of Computer Science and Information Theory, Budapest University of Technology and Economics, 1117 Budapest, Magyar tudósok körútja 2Hungary

Personalised recommendations