Half-integral vertex covers on bipartite bidirected graphs: total dual integrality and cut-rank

In this paper we study systems of the form $b\leq Mx\leq d$, $l\leq x\leq u$, where $M$ is obtained from a totally unimodular matrix with two nonzero elements per row by multiplying by 2 some of its columns, and where $b,d,l,u$ are integral vectors. We give an explicit description of a totally dual integral system that describes the integer hull of the polyhedron $P$ defined by the above inequalities. Since the inequalities of such a totally dual integral system are Chvátal inequalities for $P$, our result implies that the matrix $M$ has cut-rank 1. We also derive a strongly polynomial time algorithm to find an integral optimal solution for the dual of the problem of minimizing a linear function with integer coefficients over the aforementioned totally dual integral system.