Abstract
Given an undirected, connected graph, the aim of the minimum branch-node spanning tree problem is to find a spanning tree with the minimum number of nodes with degree larger than 2. The problem is motivated by network design problems where junctions are significantly more expensive than simple end- or through-nodes, and are thus to be avoided. Unfortunately, it is NP-hard to recognize instances that admit an objective value of zero, rendering the search for guaranteed approximation ratios futile.
We suggest to investigate a complementary formulation, called maximum path-node spanning tree, where the goal is to find a spanning tree that maximizes the number of nodes with degree at most two. While the optimal solutions (and the practical applications) of both formulations coincide, our formulation proves more suitable for approximation. In fact, it admits a trivial 1/2-approximation algorithm. Our main contribution is a local search algorithm that guarantees a ratio of 6/11.
This research started from an open problem posed at Dagstuhl Seminar 11191. The authors are grateful for Dagstuhl’s support.
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
Bhatia, R., Khuller, S., Pless, R., Sussmann, Y.J.: The full degree spanning tree problem. In: Proc. of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1999), pp. 864–865 (1999)
Cerulli, R., Gentili, M., Iossa, A.: Bounded-degree spanning tree problems: models and new algorithms. Computational Optimization and Applications 42(3), 353–370 (2009)
Fürer, M., Raghavachari, B.: Approximating the minimum degree spanning tree to within one from the optimal degree. In: Proc. of the Third Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms (SODA 1992), pp. 317–324 (1992)
Gargano, L., Hammar, M.: There are spanning spiders in dense graphs (and we know how to find them). In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 802–816. Springer, Heidelberg (2003)
Gargano, L., Hammar, M., Hell, P., Stacho, L., Vaccaro, U.: Spanning spiders and light-splitting switches. Discrete Mathematics 285(1-3), 83–95 (2004)
Gargano, L., Hell, P., Stacho, L., Vaccaro, U.: Spanning trees with bounded number of branch vertices. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 355–365. Springer, Heidelberg (2002)
Guha, S., Khuller, S.: Approximation algorithms for connected dominating sets. Algorithmica 20(4), 374–387 (1998)
Khuller, S., Bhatia, R., Pless, R.: On local search and placement of meters in networks. In: Proc. of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2000), pp. 319–328 (2000)
Knauer, M., Spoerhase, J.: Better approximation algorithms for the maximum internal spanning tree problem. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 459–470. Springer, Heidelberg (2009)
Lu, H.I., Ravi, R.: The power of local optimization: Approximation algorithms for maximum-leaf spanning tree. In: Proceedings of the Thirtieth Annual Allerton Conference on Communication, Control and Computing, pp. 533–542 (1992)
Prieto, E., Sloper, C.: Reducing to independent set structure – the case of k-internal spanning tree. Nordic Journal of Computing 12(3), 308–318 (2005)
Salamon, G., Wiener, G.: On finding spanning trees with few leaves. Information Processing Letters 105, 164–169 (2008)
Salamon, G.: Approximating the maximum internal spanning tree problem. Theoretical Computer Science 410(50), 5273–5284 (2009)
Salamon, G.: Degree-Based Spanning Tree Optimization. PhD thesis, Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Hungary (2010)
Silva, D.M., Silva, R.M.A., Mateus, G.R., Gonçalves, J.F., Resende, M.G.C., Festa, P.: An iterative refinement algorithm for the minimum branch vertices problem. In: Pardalos, P.M., Rebennack, S. (eds.) SEA 2011. LNCS, vol. 6630, pp. 421–433. Springer, Heidelberg (2011)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chimani, M., Spoerhase, J. (2013). Approximating Spanning Trees with Few Branches. In: Erlebach, T., Persiano, G. (eds) Approximation and Online Algorithms. WAOA 2012. Lecture Notes in Computer Science, vol 7846. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38016-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-38016-7_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38015-0
Online ISBN: 978-3-642-38016-7
eBook Packages: Computer ScienceComputer Science (R0)