Abstract
An undirected biconnected graph G with non negative weights on the edges is given. In the cycle space associated with G, a subspace of the vector space of G, we define as weight of a basis the maximum among the weights of the cycles of the basis. The problem we consider is that of finding a basis of minimum weight for the cycle space. It is easy to see that if we do not put additional constraints on the basis, then the problem is easy and there are fast algorithms for solving it. On the other hand if we require the basis to be fundamental, i.e. to consist of the set of all fundamental cycles of G with respect to the chords of a spanning tree of G, then we show that the problem is NP-hard and cannot be approximated within 2 .β∀β > 0, even with uniform weights, unless P=NP. We also show that the problem remains NP-hard when restricted to the class of complete graphs; in this case it cannot be approximated within 13/11 . β∀β> 0, unless P=NP; it is instead approximable within 2 in general, and within 3/2 if the triangle inequality holds.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bley A., Grotschel M. and Wessaly R., Design of Broadband Virtual Private Networks: Model and Heuristics for the B-Win, Pre-print, Konrad-Zuse-Zentrum für Informationstechnik Berlin, 1998.
Deo N., Prabhu G. M. and Krishamoorthy M. S., Algorithms for Generating Fundamental Cycles in a Graph, ACM Trans. Math. Software, 8 (1982), pp 26–42
Horton J. D., A polynomial-time algorithm to find the shortest cycle basis of a graph, SIAM J. Computing, vol. 16, n.2 (1987), pp 358–366
Knuth D. E., The Art of Computer Programming, vol.1, Addison-Wesley, Reading Mass, (1968), pp 363–368
Syslo M. M., On Cycle Bases of a Graph, Networks, vol 9 (1979) pp 123–132
Sussenouth E. Jr, A graph theoretical algorithm for matching chemical structures. J. Chem. Doc. 5,1 (1965), pp 36–43
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Galbiati, G. (2001). On Min-Max Cycle Bases. In: Eades, P., Takaoka, T. (eds) Algorithms and Computation. ISAAC 2001. Lecture Notes in Computer Science, vol 2223. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45678-3_11
Download citation
DOI: https://doi.org/10.1007/3-540-45678-3_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42985-2
Online ISBN: 978-3-540-45678-0
eBook Packages: Springer Book Archive