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Joerg Bader (talk | contribs) Clarified definition, simplified statements, added totally balanced matrix. |
Joerg Bader (talk | contribs) m →Connection to other matrix classes: Style |
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Moreover, any 0-1 matrix that is [[totally unimodular]] is also balanced. The following matrix is a balanced matrix as it does not contain any submatrix that is incidence matrix of an odd order cycle:
:<math>
1 & 1 & 1 & 1\\
1 & 1 & 0 & 0\\
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\end{bmatrix}</math>
Since this matrix is not totally unimodular (its determinant is -2), 0-1 totally unimodular matrices are a [[proper subset]] of balanced matrices. <ref name="Schrijver1998" />
For example, balanced matrices arise as the coefficient matrix in special cases of the [[set partitioning problem]]. <ref name="RyanFalkner">{{cite journal|last1 = Ryan | first1 = D. M. | last2 = Falkner | first2 = J. C. | year = 1988 | title = On the integer properties of scheduling set partitioning models | doi = 10.1016/0377-2217(88)90233-0 | journal = European Journal of Operational Research | volume = 35 | issue = 3 | pages = 442–456}}</ref>
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