Half-Integral Vertex Covers on Bipartite Bidirected Graphs: Total Dual Integrality and Cut-Rank

In this paper we study systems of the form b ≤ Mx ≤ d, l ≤ x ≤ 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 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 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.