Sequence independent lifting for a set of submodular maximization problems

We study the polyhedral structure of a mixed 0-1 set arising in the submodular maximization problem, given by \(P = \{(w,x)\in \mathbb {R}\times \{0,1\}^n: w\le f(x), x\in \mathcal {X}\}\), where submodular function f(x) is represented by a concave function composed with a linear function, and \(\mathcal {X}\) is the feasible region of binary variables x. For \(\mathcal {X}= \{0,1\}^n\), two families of facet-defining inequalities are proposed for the convex hull of P through restriction and lifting using submodular inequalities. When \(\mathcal {X}\) is a partition matroid, we propose a new class of facet-defining inequalities for the convex hull of P through multidimensional sequence independent lifting. Our results enable us to unify and generalize the existing results on valid inequalities for the mixed 0-1 knapsack. Finally, we perform some preliminary computational experiments to illustrate the superiority of our facet-defining inequalities.