Mixed-Integer Vertex Covers on Bipartite Graphs

Let A be the edge-node incidence matrix of a bipartite graph G = (U, V ;E), I be a subset the nodes of G, and b be a vector such that 2b is integral. We consider the following mixed-integer set: X(G, b, I) = {x : Ax ≥ b, x ≥ 0, xi integer for all i ∈ I}. We characterize conv(X(G, b, I)) in its original space. That is, we describe a matrix (A′, b′) such that conv(X(G, b, I)) = {x : A′x ≥ b′}. This is accomplished by computing the projection onto the space of the x-variables of an extended formulation, given in [1], for conv(X(G, b, I)). We then give a polynomial algorithm for the separation problem for conv(X(G, b, I)), thus showing that the problem of optimizing a linear function over the set X(G, b, I) is polynomially solvable.