An integer analogue of Carathéodory's theorem

AbstractWe prove a theorem on Hilbert bases analogous to Carathéodory's theorem for convex cones. The result is used to give an upper bound on the number of nonzero variables needed in optimal solutions to integer programs associated with totally dual integral systems. For integer programs arising from perfect graphs the general bounds are improved to show that if G is a perfect graph with n nodes and w is a vector of integral node weights, then there exists a minimum w-covering of the nodes that uses at most n distinct cliques