Abstract
In this paper new and generalized lower bounds for the graph partitioning problem are presented. These bounds base on the well known lower bound of embedding a clique into the given graph with minimal congestion. This is equivalent to a multicommodity flow problem where each vertex sends a commodity of size one to every other vertex. Our new bounds use arbitrary multicommodity flow instances for the bound calculation, the critical point for the lower bound is the guaranteed cut flow of the instances. Furthermore, a branch&bound procedure basing on these bounds is presented and finally it is shown that the new bounds are also useful for lower bounds on classes of graphs, e.g. the Butterfly and Benes graph.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This work was partially supported by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT), and by the German Science Foundation (DFG) project SFB-376.
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
L. Brunetta, M. Conforti, and G. Rinaldi. A branch-and-cut algorithm for the equicut problem. Mathematical Programming, 78:243–263, 1997.
C. Bornstein, A. Litman, B. Maggs, R. Sitaraman, and T. Yatzkar. On the Bisection Width and Expansion of Butterfly Networks. In Proceedings of the 1st Merged International Parallel Processing Symposium and Symposium on Parallel and Distributed Processing (IPPS/SPDP-98), pages 144–150. IEEE Computer Society, 1998.
P.-O. Fjällström. Algorithms for graph partitioning: A survey. Linköping Electronic Articles in Computer and Information Science, 1998.
L. K. Fleischer. Approximating Fractional Multicommodity Flow Independent of the Number of Commodities. SIAM Journal on Discrete Mathematics, 13(4):505–520, 2000.
C. E. Ferreira, A. Martin, C. de Souza, R. Weismantel, and L. A. Wolsey. The node capacitated graph partitioning problem: a computational study. Mathematical Programming, 81:229–256, 1998.
ILOG. CPLEX 7.0 Reference Manual, 2000.
E. Johnson, A. Mehrotra, and G. Nemhauser. Min-cut clustering. Mathematical Programming, 62:133–151, 1993.
S. E. Karisch, F. Rendl, and J. Clausen. Solving graph bisection problems with semidefinite programming. INFORMS Journal on Computing, 12(3):177–191, 2000.
F. T. Leighton. Introduction to Parallel Algorithms and Architectures. Morgan Kaufman, 1992.
T. Leighton, F. Makedon, S. Plotkin, C. Stein, E. Tardos, and S. Tragoudas. Fast Approximation Algorithms for Multicommodity Flow Problems. Journal of Computer and System Sciences, 50(2):228–243, 1995.
R. D. McBride. Progress made in solving the multicommodity flow problem. SIAM Journal on Optimization, 8:947–955, 1998.
K. Mehlhorn and S. Näher. LEDA, a library of efficient datatypes and algorithms. Technical report, University of Saarland, 1989.
R. Preis and R. Dieckmann. The PARTY Partitioning-Library User Guide-Version 1.1. SFB 376 tr-rsfb-96-024, University of Paderborn, 1996.
C. Souza, R. Keunings, L. A. Wolsey, and O. Zone. A new approach to minimising the frontwidth in finite element calculations. Computer Methods in Applied Mechanics and Engineering, 111:323–334, 1994.
F. Shahrokhi and L. Szekely. On canonical concurrent flows, crossing number and graph expansion. Combinatorics, Probability and Computing, 3:523–543, 1994.
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
Sensen, N. (2001). Lower Bounds and Exact Algorithms for the Graph Partitioning Problem Using Multicommodity Flows. In: auf der Heide, F.M. (eds) Algorithms — ESA 2001. ESA 2001. Lecture Notes in Computer Science, vol 2161. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44676-1_33
Download citation
DOI: https://doi.org/10.1007/3-540-44676-1_33
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42493-2
Online ISBN: 978-3-540-44676-7
eBook Packages: Springer Book Archive