Abstract
Given a graphG=[V, E] with positive edge weights, the max-cut problem is to find a cut inG such that the sum of the weights of the edges of this cut is as large as possible. Letg(K) be the class of graphs whose longest odd cycle is not longer than2K+1, whereK is a nonnegative integer independent of the numbern of nodes ofG. We present an O(n 4K) algorithm for the max-cut problem for graphs ing(K). The algorithm is recursive and is based on some properties of longest and longest odd cycles of graphs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Barahona, “Sur la complexité du problème du verre de spins”, Rapport de Recherche No. 177 Mathématiques Appliquées et Informatiques. Université Scientifique et Medicale de Grenoble, France, October 1979.
F. Barahona, “On the complexity of max cut”, Rapport de Recherche No. 186. Mathématiques Appliquées et Informatiques, Université Scientifique et Medicale de Grenoble, France February 1980.
F. Barahona, “Balancing signed graphs of fixed genus in polynomial time”. Depto. de Matematicas, Universidad de Chile, Santiago, Chile June, 1981.
P. Erdös and H. Hajnal, “On chromatic number of graphs and set systems”,Acta Mathematica Academiae Scientiarum Hungaricae 17 (1966) 61–69.
M.R. Garey and D.S. Johnson.Computers and intractability: A guide to the theory of NP-completeness (Freeman, San Francisco, 1979).
M. Grötschel and W.R. Pulleyblank, “Weakly bipartite graphs and the max-cut problem”,Operations Research Letters 1 (1981) 23–27.
F.O. Hadlock, “Finding a maximum cut of a planar graph in polynomial time”,SIAM Journal on Computing 4 (1975) 221–225.
W.L. Hsu, Y. Ikura and G.L. Nemhauser, “A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles”,Mathematical Programming 20 (1981) 225–232.
G.I. Orlova and Y.G. Dorfman, “Finding the maximum cut in a planar graph”,Engineering Cybernetics 10 (1975) 502–506.
R.E. Tarjan, “Depth-first search and linear graph algorithms”,SIAM Journal on Computing 1 (1972) 146–159.
H.-J. Voss, “Graphs with prescribed maximal subgraphs and critical chromatic graphs”,Commentationes Mathematicae Universitatis Carolinae 18 (1977) 129–142.
M. Yannakakis, “Node- and edge-deletion NP-complete problems”,Proc. 10th Annual ACM Symposium on Theory of Computing, Association of Computing Machinery (New York, 1978) pp. 253–264.
Author information
Authors and Affiliations
Additional information
This research was supported by National Science Foundation Grant ECS-8005350 to Cornell University.
Rights and permissions
About this article
Cite this article
Grötschel, M., Nemhauser, G.L. A polynomial algorithm for the max-cut problem on graphs without long odd cycles. Mathematical Programming 29, 28–40 (1984). https://doi.org/10.1007/BF02591727
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02591727