The submodular knapsack polytope

The submodular knapsack set is the discrete lower level set of a submodular function. The modular case reduces to the classical linear 0–1 knapsack set. One motivation for studying the submodular knapsack polytope is to address 0–1 programming problems with uncertain coefficients. Under various assumptions, a probabilistic constraint on 0–1 variables can be modeled as a submodular knapsack set.In this paper we describe cover inequalities for the submodular knapsack set and investigate their lifting problem. Each lifting problem is itself an optimization problem over a submodular knapsack set. We give sequence-independent upper and lower bounds on the valid lifting coefficients and show that whereas the upper bound can be computed in polynomial time, the lower bound problem is NPNP-hard. Furthermore, we present polynomial algorithms based on parametric linear programming and computational results for the conic quadratic 0–1 knapsack case.