Total unimodularity and the transportation problem: a generalization
نویسندگان
چکیده
منابع مشابه
Generalisations of total unimodularity
In this paper we examine possible generalisations of total unimodularity. To this end, we introduce two concepts: total k-modularity and k-regularity. Total k-modularity extends the permitted values for the subdeterminants of a matrix to the powers of k, while k-regularity sets requirements on the inverses of non-singular submatrices. It is shown that the advantageous properties of totally unim...
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Definition 1.1. A matrix A ∈ Zm×n is totally unimodular if the determinant of every square submatrix B ∈ Zk×k equals either −1, 0, or +1. Alternatively, by Cramer’s rule, A ∈ Zm×n is totally unimodular if every nonsingular submatrix B ∈ Zk×k has an integral inverse B−1 ∈ Zk×k. Recall: B−1 = 1 det B B ∗, where B∗ is the adjugate matrix (transpose of the matrix of cofactors) of B. One important c...
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Given a simplicial complex K with weights on its simplices and a chain on it, the Optimal Homologous Chain Problem (OHCP) is to find a chain with minimal weight that is homologous (over Z) to the given chain. The OHCP is NP-complete, but if the boundary matrix of K is totally unimodular (TU), it becomes solvable in polynomial time when modeled as a linear program (LP). We define a condition on ...
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ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 1974
ISSN: 0024-3795
DOI: 10.1016/0024-3795(74)90003-2