Abstract
We consider the problem of computing a minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over \(\mathbb{F}_{2}\) generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Minimum cycle basis are useful in a number of contexts, e.g. the analysis of electrical networks and structural engineering.
The previous best algorithm for computing a minimum cycle basis has running time O(m ω n), where ω is the best exponent of matrix multiplication. It is presently known that ω<2.376. We exhibit an O(m 2 n+mn 2log n) algorithm. When the edge weights are integers, we have an O(m 2 n) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(m ω) time. For any ε>0, we also design an 1+ε approximation algorithm. The running time of this algorithm is O((m ω/ε)log (W/ε)) for reasonably dense graphs, where W is the largest edge weight.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Cassell, A.C., Henderson, J.C., Ramachandran, K.: Cycle bases of minimal measure for the structural analysis of skeletal structures by the flexibility method. Proc. R. Soc. Lond. Ser. A 350, 61–70 (1976)
Chua, L.O., Chen, L.: On optimally sparse cycle and coboundary basis for a linear graph. IEEE Trans. Circuit Theory 20, 495–503 (1973)
Cohen, E., Zwick, U.: All-pairs small-stretch paths. J. Algorithms 38, 335–353 (2001)
Coppersmith, D., Winograd, S.: Matrix multiplications via arithmetic progressions. J. Symb. Comput. 9, 251–280 (1990)
de Pina, J.C.: Applications of shortest path methods. PhD thesis, University of Amsterdam, Netherlands (1995)
Deo, N., Prabhu, G.M., Krishnamoorthy, M.S.: Algorithms for generating fundamental cycles in a graph. ACM Trans. Math. Softw. 8, 26–42 (1982)
Galil, Z., Margalit, O.: All pairs shortest paths for graphs with small integer length edges. J. Comput. Syst. Sci. 54, 243–254 (1997)
Golynski, A., Horton, J.D.: A polynomial time algorithm to find the minimum cycle basis of a regular matroid. In: 8th Scandinavian Workshop on Algorithm Theory (2002)
Hartvigsen, D.: Minimum path bases. J. Algorithms 15(1), 125–142 (1993)
Hartvigsen, D., Mardon, R.: When do short cycles generate the cycle space? J. Comb. Theory Ser. B 57, 88–99 (1993)
Hartvigsen, D., Mardon, R.: The all-pairs min cut problem and the minimum cycle basis problem on planar graphs. J. Discret. Math. 7(3), 403–418 (1994)
Horton, J.D.: A polynomial-time algorithm to find a shortest cycle basis of a graph. SIAM J. Comput. 16, 359–366 (1987)
Hubicka, E., Syslo, M.M.: Minimal bases of cycles of a graph. Recent Adv. Graph. Theory, 283–293 (1975)
Kavitha, T., Mehlhorn, K., Michail, D.: New approximation algorithms for minimum cycle bases of graphs. In: Proceedings of 24th International Symposium on Theoretical Aspects of Computer Science (STACS) (2007)
Kolasinska, E.: On a minimum cycle basis of a graph. Zastos. Mat. 16, 631–639 (1980)
Mehlhorn, K., Michail, D.: Implementing minimum cycle basis algorithms. In: Proceedings of 4th International Workshop on Experimental and Efficient Algorithms. Lecture Notes in Computer Science, vol. 3503, pp. 32–43. Springer, New York (2005)
Padberg, M.W., Rao, M.R.: Odd minimum cut-sets and b-matchings. Math. Oper. Res. 7, 67–80 (1982)
Pettie, S., Ramachandran, V.: Computing shortest paths with comparisons and additions. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 267–276 (2002)
Seidel, R.: On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci. 51, 400–403 (1995)
Stepanec, G.F.: Basis systems of vector cycles with extremal properties in graphs. Uspekhi Mat. Nauk. 19, 171–175 (1964)
Tewari, G., Gotsman, C., Gortler, S.J.: Meshing genus-1 point clouds using discrete one-forms. Comput. Graph. 30(6), 917–926 (2006)
Thorup, M.: Undirected single-source shortest paths with positive integer weights in linear time. J. ACM 46, 362–394 (1999)
Thorup, M.: Floats, integers, and single source shortest paths. J. Algorithms 35, 189–201 (2000)
Zwick, U.: All pairs shortest paths in weighted directed graphs—exact and approximate algorithms. In: Proceedings of the 39th Annual IEEE FOCS, pp. 310–319 (1998)
Zykov, A.A.: Theory of Finite Graphs. Nauka, Novosibirsk (1969)
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of this paper appeared in Kavitha et al. (31st International Colloquium on Automata, Languages and Programming (ICALP), pp. 846–857, 2004).
T. Kavitha and K.E. Paluch were in Max-Planck-Institut für Informatik, Saarbrücken, Germany, while this work was done.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Kavitha, T., Mehlhorn, K., Michail, D. et al. An \(\tilde{O}(m^{2}n)\) Algorithm for Minimum Cycle Basis of Graphs. Algorithmica 52, 333–349 (2008). https://doi.org/10.1007/s00453-007-9064-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-007-9064-z