Abstract
Ani-j xcut of a setV={1, ...,n} is defined to be a partition ofV into two disjoint nonempty subsets such that bothi andj are contained in the same subset. When partitions are associated with costs, we define thei-j xcut problem to be the problem of computing ani-j xcut of minimum cost. This paper contains a proof that the\((\begin{array}{*{20}c} n \\ 2 \\ \end{array} )\) minimum xcut problems have at mostn distinct optimal solution values. These solutions can be compactly represented by a set ofn partitions in such a way that the optimal solution to any of the problems can be found inO(n) time. For a special additive cost function that naturally arises in connection to graphs, some interesting properties of the set of optimal solutions that lead to a very simple algorithm are presented.
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References
C.K. Cheng and T.C. Hu, Maximum concurrent flow and minimum ratio cut, Technical Report CS88-141, Department of Computer Science and Engineering, University of California, San Diego (1988).
M. Conforti and M.R. Rao, Some new matroids on graphs: Cut sets and the max cut problem, Math. Oper. Res. 12(1987)193–204.
R.E. Gomory and T.C. Hu, Multi-terminal network flows, J. SIAM 9(1961)551–570.
F. Granot and R. Hassin, Multi-terminal maximum flows in node capacitated networks, Discr. Appl. Math. 13(1986)157–163.
D. Gusfield and D. Naor, Extracting maximal information on sets of minimum cuts, Technical Report, University of California, Davis (1988).
R. Hassin, Solution bases of multiterminal cut problems, Math. Oper. Res. 13(1988)535–542.
R. Hassin, An algorithm for computing maximum solution bases, Oper. Res. Lett. 9(1990) 315–318.
H. Nagamochi and T. Ibaraki,Computing Edge-Connectivity in Multiple and Capacitated Graphs, Algorithms, Lecture Notes in Computer Science 45(1990), pp. 12–20.
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Hassin, R. Multiterminal xcut problems. Ann Oper Res 33, 215–225 (1991). https://doi.org/10.1007/BF02115756
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DOI: https://doi.org/10.1007/BF02115756