Abstract
We study the approximability of some problems which aim at finding spanning trees in undirected graphs which maximize, instead of minimizing, a single objective function representing a form of benefit or usefulness of the tree. We prove that the problem of finding a spanning tree which maximizes the number of paths which connect pairs of vertices and pass through a common arc can be polynomially approximated within a factor of 1.5. It is known that this problem can be solved exactly in polynomial time if the graph is 2-connected [14]; we extend this result to graphs having at most two articulation points. We leave open whether in the general case the problem admits a polynomial time approximation scheme or is MAX-SNP hard and therefore not polynomially approximable whithin any constant 1+∈, ∈>0, unless P=NP. On the other hand we show that the problems of finding a spanning tree which has maximum diameter, or maximum height with respect to a specified root, or maximum sum of the distances between all pairs of vertices, or maximum sum of the distances from a specified root to all remaining vertices, are not polynomially approximable within any constant factor, unless P=NP. The same result holds for the problem of finding a lineal spanning tree with maximum height, and this solves a problem which was left open in [6].
Partially supported by the M.U.R.S.T. 40% Research Project “Algoritmi, Modelli di Calcolo e Strutture Informative”.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Arora, S., Lund, C., Motwani, R., Sudan, M., and Szegedyr, M.: Proof verification and hardness of approximation problems. Proc. 33-rd Ann. IEEE Symp. on Foundations of Computer Science (1992) 14–23
Camerini, P., Galbiati, G., and Maffioli, F.: Complexity of spanning tree problems: Part I. European Journal of Operational Research 5 (1980) 346–352
Camerini, P., Galbiati, G., and Maffioli, F.: On the complexity of finding multi-constrained spanning trees. Discrete Applied Mathematics 5 (1983) 39–50
Camerini, P., Galbiati, G., and Maffioli, F.: The complexity of weighted multi-constrained spanning tree problems. Colloquia Mathematica Societatis Janos Bolyai 44 (1986) 53–101
Crescenzi, P., Panconesi, A.: Completeness in approximation classes. Information and Computation 2 (1993) 241–262
Fellows, M.R., Friesen, D.K., and Langston, M.A.: On Finding Optimal and Near-Optimal Lineal Spanning Trees. Algorithmica 3 (1988) 549–560
Furer, M., Raghavachari, B.: Approximating the Minimum Degree Spanning Tree to within One from the Optimal Degree. Proc. of the 3rd Annual ACM-SIAM Symposium on Discrete Algorithms (1992) 317–324
Galbiati, G., Maffioli, F., and Morzenti, A.: A Short Note on The Approximability of The Maximum Leaves Spanning Tree Problem. Information Processing Letters 52 (1994) 45–49
Goemans, M.X., Goldberg, A.V., Plotkin, S., Dhmoys, D.B., Tardos, E., and Williamson, D.P.: Improved Approximation Algorithms for Network Design Problems. Proc. of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms (1994) 223–232
Lu, Hsueh-I, and Ravi, R.: The Power of Local Optimization: Approximation Algorithms for Maximum-Leaf Spanning Tree. Proc. Allerton Conference (1992) 533–542
Karger, D., Motwani, R., and Rankumar, G.D.S.: On Approximating the Longest Path in a Graph. Proc. of the 3rd Workshop on Algorithms and Data Structures (Lectures Notes in Comput. Sci. vol. 709) Springer-Verlag (1993) 421–432
Khuller, S., Raghavachari, B., and Young, N.: Balancing Minimum Spanning Trees and Shortes-Path Trees. Proc. of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms (1993)
Khuller, S., Raghavachari, B., and Young, N.: Low Degree Spanning Trees of Small Weight. Proc. 26th Annual Symp. on the Theory of Computing (STOC '94) (1994)
Lovász, L.: Combinatorial Problems and Exercises. North-Holland (1979)
Papadimitriou, C.H., and Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. and Syst. Sci. 43 (1991) 425–440
Ravi, R., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., and Hunt, H.B., III: Many birds with one stone: multi-objective approximation algorithms. Proc. 25th Annual Symp. on The Theory of Computing (STOC '94) (1993) 438–447
Ravi, R., Sundaram, R., Marathe, M.V., Rosenkrantz, D.J., and Ravi, S.S.: Spanning Trees Short or Small. Proc. of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms (1994) 546–555
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Galbiati, G., Morzenti, A., Maffioli, F. (1995). On the approximability of some maximum spanning tree problems. In: Baeza-Yates, R., Goles, E., Poblete, P.V. (eds) LATIN '95: Theoretical Informatics. LATIN 1995. Lecture Notes in Computer Science, vol 911. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59175-3_97
Download citation
DOI: https://doi.org/10.1007/3-540-59175-3_97
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59175-7
Online ISBN: 978-3-540-49220-7
eBook Packages: Springer Book Archive