Valid inequalities, covering problems and discrete dynamic programs

Various discrete optimization problems such as the integer and 0–1 programming problems, and the travelling salesman problem have been represented as discrete dynamic programming, or network problems. We show how such representations lead naturally to a characterization of the valid inequalities for the feasible solution sets Q of such problems. In particular we obtain polytopes Γ of valid inequalities having the facets of Q among their extreme points. In addition the problems of “packing” or “covering” with feasible solutions to the discrete problem have natural network representations, which are the duals of problems over Γ. Reversing the approach, any special properties of the valid inequalities can in turn be used to give new formulations of the corresponding network problems. In particular this allows a reformulation of the “minimum equivalent knapsack inequality” problem, and the “cutting stock” problem.