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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Salamon, G., Wiener, G.: On finding spanning trees with few leaves (submitted 2006)
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)
Schrijver, A.: 50: Shortest spanning trees. In: Combinatorial optimization. In: Polyhedra and efficiency, vol. B, pp. 855–876. Springer, Heidelberg (2003)
Zhang, S., Wang, Z.: Scattering number in graphs. Networks 37, 102–106 (2001)
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)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Salamon, G. (2007). Approximation Algorithms for the Maximum Internal Spanning Tree Problem. In: Kučera, L., Kučera, A. (eds) Mathematical Foundations of Computer Science 2007. MFCS 2007. Lecture Notes in Computer Science, vol 4708. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74456-6_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-74456-6_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74455-9
Online ISBN: 978-3-540-74456-6
eBook Packages: Computer ScienceComputer Science (R0)