Abstract
Let A be a (0,1)-matrix with n rows and m columns, considered as the incidence matrix of a hypergraph H with edges E 1, E 2, …, E n (the columns) and with vertices x 1, x 2, …, x n (the rows). H is called a balanced hypergraph if A does not contain a square sub-matrix of odd order with exactly two ones in each row and in each column. In this paper, we prove more “minimax” equalities for balanced hypergraphs, than those already proved in Berge [1], Berge and Las Vergnas [3], Fulkerson et al. [7], Lovász [12]; in fact, the known results will follow easily from our main theorem.
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References
C. Berge, “Balanced matrices”, Mathematical Programming 2 (1972) 19–31.
C. Berge, “Sur une extension de la théorie des matrices bi-stochastiques”, in: Dall'Aglio, ed., Studi de probabilita, satistitica e ricerca operativa, in onore di G. Pompilj. (Oderisi-Gubbio, Rome 1970) 475–482.
C. Berge and M. Las Vergnas, “Sur un théorème du type König pour hypergraphes”, Annals of the New York Academy of Sciences 175 (1970) 32–40.
V. Chvátal, “On certain polytopes associated with graphs”, Journal of Combinatorial Theory 18(B) (1975) 138–154.
V. Chvátal, “On the strong perfect graph conjecture”, Journal of Combinatorial Theory 20(B) (1976) 139–141.
D.R. Fulkerson, “Blocking and anti blocking pairs of polyhedra”, Mathematical Programming 1 (1971) 168–194.
D.R. Fulkerson, A.J. Hoffman and R. Oppenheim, “On balanced matrices”, Mathematical Programming Study 1 (1974) 120–132.
R.P. Gupta, “An edge-coloration theorem for bipartite graphs of paths in trees”, Discrete Mathematics 23 (1978) 229–233.
A.J. Hoffman, “A generalisation of max-flow min-cut”, Mathematical Programming 6 (1974) 352–359.
A. Lehman, “On the width-length inequality”, mimeo (1965).
L. Lovasz, “Normal hypergraphs and perfect graph conjecture”, Discrete Mathematics 2 (1972) 253–267.
L. Lovász, “On two mini-max theorems in graphs”, Journal of Combinatorial Theory 21(B) (1976) 96–103.
M.W. Padberg, “On the facial structure of set packing polyhedra”, Mathematical Programming 5 (1973) 199–215.
M.W. Padberg, “Perfect zero-one matrices”, Mathematical Programming 6 (1974) 180–196.
M.W. Padberg, “Almost integral polyhedra related to certain combinatorial optimization problems”, Linear Algebra and Appl. 15 (1976) 69–88.
M.W. Padberg, “Almost integral polyhedra related to certain combinatorial optimization problems”, Linear Algebra and its Applications, 15 (1976) 69–88.
A. Schrijver, Fractional packing and covering, in: Packing and covering, (Mathematisch Centrum, Amsterdam, 1978) 175–248.
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© 1980 The Mathematical Programming Society
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Berge, C. (1980). Balanced matrices and property (G). In: Padberg, M.W. (eds) Combinatorial Optimization. Mathematical Programming Studies, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120894
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DOI: https://doi.org/10.1007/BFb0120894
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Publisher Name: Springer, Berlin, Heidelberg
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