Strong unimodularity for matrices and hypergraphs

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Abstract

A 0-1 matrix A is called strongly unimodular if all the bases of (A, I) are triangular. We develop equivalent conditions for strong unimodularity, first in algebraic, then in graph theoretic terms. This provides a link with the theory of unimodular and balanced hypergraphs, and allows us to produce a polynomial-time recognition algorithm for strongly unimodular matrices. We consider next the constraint matrix of the problem obtained by linearizing a general, unconstrained optimization problem in 0-1 variables. Because that matrix has 0, 1 and -1 entries, we are led to introduce the concept of signed hypergraph in which every edge is affected of a positive or negative sign. Our results on strong unimodularity are extended to the class of signed hypergraphs. © 1986.

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Crama, Y., Hammer, P. L., & Ibaraki, T. (1986). Strong unimodularity for matrices and hypergraphs. Discrete Applied Mathematics, 15(2–3), 221–239. https://doi.org/10.1016/0166-218X(86)90044-2

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