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 of the nodes of G, and b be a vector such that 2b is integral. We consider the following mixed-integer set: We characterize conv(X(G,b,I)) in its original space. That is, we describe a matrix (C,d) such that conv(X(G,b,I)) = {x : Cx ≥ d}. 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-time 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 solvable in polynomial time.