Combinatorial optimization games form an important subclass of cooperative games. In recent years, increased attention has been given to the issue of finding good cost shares for such games. In this paper, we define a very general class of games, called integer minimization games, which includes the combinatorial optimization games in the literature as special cases. We then present new techniques, based on row and column generation, for computing good cost shares for these games. To illustrate the power of these techniques, we apply them to traveling salesman and vehicle routing games. Our results generalize and unify several results in the literature. The main underlying idea is that suitable valid inequalities for the associated combinatorial optimization problems can be used to derive improved cost shares.