Abstract
A balanced graph partition on a vertex-weighted graph is a partition of the vertex set such that the partition has k parts and the disparity, which is defined as the ratio of the maximum total weight of parts to the minimum one, is at most r. In this paper, a novel algorithm is proposed that enumerates all the graph partitions with small disparity. Experimental results show that five millions of partitions with small disparity for some graph with more than 100 edges can be enumerated within ten minutes.
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 subscriptionsReferences
Andreev, K., Räcke, H.: Balanced graph partitioning. Theor. Comput. Syst. 39(6), 929–939 (2006)
Flajolet, P., Noy, M.: Analytic combinatorics of non-crossing configurations. Discrete Math. 204(1), 203–229 (1999)
Hardy, G., Lucet, C., Limnios, N.: K-terminal network reliability measures with binary decision diagrams. IEEE Trans. Reliab. 56(3), 506–515 (2007)
Imai, H., Sekine, K., Imai, K.: Computational investigations of all-terminal network reliability via BDDs. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E82–A, 714–721 (1999)
OEIS Foundation Inc.: The on-line encyclopedia of integer sequences (2011). http://oeis.org/A054726
Inoue, T., Takano, K., Watanabe, T., Kawahara, J., Yoshinaka, R., Kishimoto, A., Tsuda, K., Minato, S., Hayashi, Y.: Distribution loss minimization with guaranteed error bound. IEEE Trans. Smart Grid 5(1), 102–111 (2014)
Karpinski, M.: Approximability of the minimum bisection problem: an algorithmic challenge. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 59–67. Springer, Heidelberg (2002). doi:10.1007/3-540-45687-2_4
Kawahara, J., Inoue, T., Iwashita, H., Minato, S.: Frontier-based search for enumerating all constrained subgraphs with compressed representation. Hokkaido University, Division of Computer Science, TCS Technical reports TCS-TR-A-13-76 (2014)
King, D.M., Jacobson, S.H., Sewell, E.C.: Efficient geo-graph contiguity and hole algorithms for geographic zoning and dynamic plane graph partitioning. Math. Program. 149(1–2), 425–457 (2015). http://dx.doi.org/10.1007/s10107-014-0762-4
Knuth, D.E.: The Art of Computer Programming, Vol. 4A. Combinatorial Algorithms, Part 1, 1st edn. Addison-Wesley Professional, Boston (2011)
Minato, S.: Zero-suppressed BDDs for set manipulation in combinatorial problems. In: Proceedings of the 30th ACM/IEEE Design Automation Conference, pp. 272–277 (1993)
Nemoto, T., Hotta, K.: On the limits of the reduction in population disparity between single-member election districts in Japan. Jpn. J. Electoral Stud. 20, 136–147 (2005). (in Japanese)
Sekine, K., Imai, H., Tani, S.: Computing the Tutte polynomial of a graph of moderate size. In: Staples, J., Eades, P., Katoh, N., Moffat, A. (eds.) ISAAC 1995. LNCS, vol. 1004, pp. 224–233. Springer, Heidelberg (1995). doi:10.1007/BFb0015427
Takizawa, A., Takechi, Y., Ohta, A., Katoh, N., Inoue, T., Horiyama, T., Kawahara, J., Minato, S.: Enumeration of region partitioning for evacuation planning based on ZDD. In: Proceedings of 11th International Symposium on Operations Research and its Applications in Engineering, Technology and Management (ISORA 2013), pp. 1–8 (2013)
Yoshinaka, R., Saitoh, T., Kawahara, J., Tsuruma, K., Iwashita, H., Minato, S.: Finding all solutions and instances of numberlink and slitherlink by ZDDs. Algorithms 5(2), 176–213 (2012). http://www.mdpi.com/1999-4893/5/2/176/
Acknowledgment
The authors would like to thank Dr. Toshiki Saitoh, Dr. Norihito Yasuda and Dr. Ryo Yoshinaka for their valuable comments. This work was partly supported by JSPS KAKENHI 15H05711, 24106007 and 15K00008.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Kawahara, J., Horiyama, T., Hotta, K., Minato, Si. (2017). Generating All Patterns of Graph Partitions Within a Disparity Bound. In: Poon, SH., Rahman, M., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2017. Lecture Notes in Computer Science(), vol 10167. Springer, Cham. https://doi.org/10.1007/978-3-319-53925-6_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-53925-6_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-53924-9
Online ISBN: 978-3-319-53925-6
eBook Packages: Computer ScienceComputer Science (R0)