Integer Knapsack and Flow Covers with Divisible Coefficients: Polyhedra, optimization and Separation

Three regions arising as surrogates in certain network design problems are the knapsack set X = {x [ belong ] [Z^n_+] : [ sum^n_j=1] C_jx_j ≥ b}, the simple capacitated flow set Y = {(y, x) [belong ] [R^1_+] x [Z^n_+] : y ≤ b,y ≤ [ sum^n_j=1] C_jx_j} and the set Z = {(y, x) [ belong ] [R^n_+] x [Z^n_+] : [ sum^n_j=1] y_i ≤ b, y_i ≤ C_jx_j for j = 1,...,n}, where the capacity C_(j+1) is an integer multiple Cj for all j. We present algorithms for optimization over the sets X and Y, as well as different descriptions of the convex hulls and fast combinatorial algorithms for separation. Some partial results are given for the set Z and another extensions.