Abstract
We describe a decomposition framework and a column generation scheme for solving a min-cut clustering problem. The subproblem to generate additional columns is itself an NP-hard mixed integer programming problem. We discuss strong valid inequalities for the subproblem and describe some efficient solution strategies. Computational results on compiler construction problems are reported.
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References
F. Barahona and R. Majhoub, “On the cut polytope,”Mathematical Programming 36 (1986) 157–173.
S. Chopra, “The graph partitioning polytope on series-parallel and 4-wheel free graphs,” J.L. Kellogg Graduate School of Management, Northwestern University (Evanston, IL, 1991).
S. Chopra and M.R. Rao, “The partition problem,” J.L. Kellogg Graduate School of Management, Northwestern University (Evanston, IL, 1990).
S. Chopra and M.R. Rao, “Facets of thek-partition polytope,” J.L. Kellogg Graduate School of Management, Northwestern University (Evanston, IL, 1991).
M. Conforti, M.R. Rao and A. Sassano, “The equipartition polytope. I: Formulations, dimension and basic facets,”Mathematical Programming 49 (1990) 49–70.
M. Conforti, M.R. Rao and A. Sassano, “The equipartition polytope. II: Valid inequalities, and facets,”Mathematical Programming 49 (1990) 71–90.
E. Dahlaus, D.S. Johnson, C.H. Papadimitriou, P. Seymour and M. Yanakakis, “The complexity of multiway cuts,” extended abstract (1983).
M.R. Garey and D.S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979).
O. Goldschmidt and D.S. Hochbaum, “An O(|V|2) algorithm for thek-cut problem,”Proceedings 29th Annual Symposium on Foundations of Computer Science (1985) pp. 444–451.
M. Grötschel and Y. Wakabayashi, “A cutting plane algorithm for a clustering problem,”Mathematical Programming (Series B) 45 (1989) 59–96.
M. Grötschel and Y. Wakabayashi, “Facets of the clique partitioning polytope,”Mathematical Programming 47 (1990) 367–387.
E.L. Johnson, “Modeling and strong linear programs for mixed integer programming,”NATO ASI Series, F51, Algorithms and Model Formulations in Mathematical Programming (Springer, Berlin, 1989).
A. Mehrotra, “Constrained graph partitioning: decomposition, polyhedral structure and algorithms,” PhD Dissertation, School of Industrial and Systems Engineering, Georgia Institute of Technology (Atlanta, GA, 1992).
G.L. Nemhauser and S. Park, “A polyhedral approach to edge coloring,”Operations Research Letters 10 (1991) 315–322.
G.L. Nemhauser and L.A. Wolsey,Integer and Combinatorial Optimization (Wiley, New York, 1988).
M. Padberg, “The Boolean quadratic polytope: Some characteristics, facets and relatives,”Mathematical Programming (Series B) 45 (1989) 139–172.
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This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.
This research was supported by NSF grants DMS-8719128 and DDM-9115768, and by an IBM grant to the Computational Optimization Center, Georgia Institute of Technology.
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Johnson, E.L., Mehrotra, A. & Nemhauser, G.L. Min-cut clustering. Mathematical Programming 62, 133–151 (1993). https://doi.org/10.1007/BF01585164
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DOI: https://doi.org/10.1007/BF01585164