Rational and integral k-regular matrices

Optimization with binet matrices

“…They also showed that a binet matrix is 2-regular, i.e., the inverse of any of it's non-singular submatrix is half-integral. This fact, together with a theorem proved in [3], implies that if the constraint matrix of a linear program is a binet matrix, then all basic optimal solutions are half-integral. However, the question of how to find an optimal solution in an efficient way was left open.…”

“…In [3] the authors proved that if A is an integral 2-regular matrix of size m × n, then for polyhedron Q = {x | Ax ≤ b, x ≥ 0} with integral b, the rank-1 closure Q 1 can be achieved by only half-integral cuts, i.e., valid inequalities of the form λAx ≤ bλbc where λ ∈ {0, ½ } m and λA is integral. In a compact form, this result states that Q 1 = Q ½ for integral 2-regular matrices and any integral right hand side vector, if we define Q ½ as the {0, ½ }-closure of Q, i.e., the intersection of Q with the half-spaces induced by all the possible half-integral cuts.…”

“…Appa and Kotnyek generalized network matrices in [2,3] by introducing binet matrices. A binet matrix is defined on a bidirected graph in a way similar to how a network matrix is defined on a directed graph.…”

A bidirected generalization of network matrices

“…The fundamental circuit induced by s 2 is a handcuff {s 2 , r 6 , r 4 , r 2 , r 1 , r 3 , r 4 , r 5 }, in which r 4 is traversed twice, so the final value standing in row r 4 and column s 2 is the sum of the two preliminary values of r 4 .…”

“…Our way of proving these results follows [4], in which we established the connection between k-regular matrices and integral polyhedra. Here, we deal only with the special case of 1-regular and 2-regular matrices.…”