Algorithms for a class of Min-Cut and Max-Cut problem
The k-Min-Cut (k-Max-Cut) problem consists of partitioning the vertices of an edge weighted (undirected) graph into k sets so as to minimize (maximize) the sum of the weights of the edges joining vertices in different subsets. We concentrate on the k-Max-Cut and k-Min-Cut problems defined over complete graphs that satisfy the triangle inequality, as well as on d-dimensional graphs. For the one-dimensional version of our partitioning problems, we present efficient algorithms for their solution as well as lower bounds for the time required to find an optimal solution, and for the time required to verify that a solution is an optimal one. We also establish a bound for the objective function value of an optimal solution to the k-Min-Cut and k-Max-Cut problems whose graph satisfies the triangle inequality. The existence of this bound is important because it implies that any feasible solution is a near-optimal approximation to such versions of the k-Max-Cut and k-Min-Cut problems.
KeywordsTriangle Inequality Time Algorithm Optimal Partition Linear Time Algorithm Verification Problem
Unable to display preview. Download preview PDF.
- 1.M. Ben-Or, “Lower Bounds for Algebraic Computation Trees,” Proc. 15th ACM Annual Symposium on Theory of Computing 80–86, 1983.Google Scholar
- 3.M. R. Garey, D. S. Johnson, and L. Stockmeyer, “Some Simplified NP-Complete Graph Problems,” Theoretical Computer Science, (1), 237–267, 1976.Google Scholar
- 4.M. R. Garey and D. S. Johnson, “Computers and Intractability,” Freeman, 1980.Google Scholar
- 6.J. R. Gilbert and E. Zmijewski, “Combinatorial Sparse Matrix Theory,” Chapter 5, Lecture Notes in Computer Science, UCSB, 1991.Google Scholar
- 7.T. F. Gonzalez, “Clustering to Minimize the Maximum Intercluster Distance,” Theoretical Computer Science, 38, 293–306, 1985.Google Scholar
- 8.E. Horowitz and S. Sahni, “Fundamentals of Computer Algorithms,” Computer Science Press, 1978.Google Scholar
- 9.B. W. Kernighan and S. Lin, “An Efficient Heuristic Procedure for Partitioning Graphs,” The Bell System Technical Journal, 49, 291–307, 1970.Google Scholar
- 10.B. Krishnamurthy, “An improved min-cut algorithm for partition VLSI networks,” IEEE Transactions on Computers, C(33), 438–446, 1984.Google Scholar
- 11.T. Murayama, “Algorithmic Issues for the Min-Cut and Max-Cut Problems,” M.S. Thesis, Department of Computer Science, The University of California, Santa Barbara, December 1991.Google Scholar
- 12.D. F. Wong, H. W. Leong, and C. L. Liu, “Simulated Annealing for VLSI Design,” Kluwer Academic Publishers, 1988.Google Scholar