Abstract
LetG=(V, E) be an undirected graph andA⊆V. We consider the problem of finding a minimum cost set of edges whose deletion separates every pair of nodes inA. We consider two extended formulations using both node and edge variables. An edge variable formulation has previously been considered for this problem (Chopra and Rao (1991), Cunningham (1991)). We show that the LP-relaxations of the extended formulations are stronger than the LP-relaxation of the edge variable formulation (even with an extra class of valid inequalities added). This is interesting because, while the LP-relaxations of the extended formulations can be solved in polynomial time, the LP-relaxation of the edge variable formulation cannot. We also give a class of valid inequalities for one of the extended formulations. Computational results using the extended formulations are performed.
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References
E. Balas and W.R. Pulleyblank, “The perfectly matchable subgraph polytope of a bipartite graph,”Networks 13 (1983), 495–516.
S. Chopra, “The A-cut polyhedron on series-parallel graphs,” Preprint, Northwestern University, Evanston, IL (1993).
S. Chopra and J.H. Owen, “Extended formulations for the A-cut problem,” Industrial Engineering and Management Sciences Report TR-95-66. Northwestern University, Evanston, IL (1995).
S. Chopra and M.R. Rao, “On the multiway cut polyhedron,”Networks 21 (1991) 51–89.
W.H. Cunningham, “The optimal multiterminal cut problem,” in: DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol 5 (American Mathematical Society, Providence, RI, 1991) pp. 105–120.
E. Dalhaus, D. Johnson, C. Papadimitriou, P. Seymour and M. Yannakakis, “The complexity of multiway cuts,” unpublished abstract, (1983).
O. Goldschmidt and D.S. Hochbaum, “A polynomial algorithm for thek cut problem for fixedk,”Mathematics of Operations Research 19 (1994) 24–37.
M. Grötschel, L. Lovasz and A. Schrijiver, “The ellipsoid method and its consequences in combinatorial optimization,”Combinatorica 1 (1981) 169–197.
D. Hartvigsen, “The planar multiterminal cut problem,” preprint, University of Notre Dame, Cleveland, OH (1993).
M.W. Padberg and G. Rinaldi, “A branch-and-cut algorithm for the resultion of large-scale symmetric travelling salesman problems,” Preprint, New York University (1989).
W.R. Pulleyblank, “Polyhedral combinatorics,” in: G.L. Nemhauser, A.H.G. Rinnoosy Kan and M.J. Todd, eds.,Optimization, Handbooks in Operations Research and Management Science, Vol. 1 (Amsterdam, North-Holland, 1989) pp. 371–446.
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Chopra, S., Owen, J.H. Extended formulations for the A-cut problem. Mathematical Programming 73, 7–30 (1996). https://doi.org/10.1007/BF02592096
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DOI: https://doi.org/10.1007/BF02592096