Lifting for the integer knapsack cover polyhedron

We consider the integer knapsack cover polyhedron which is the convex hull of the set consisting of n-dimensional nonnegative integer vectors that satisfy one linear constraint. We study the sequentially lifted (SL) inequality, derived by the sequential lifting from a seed inequality containing a single variable, and provide bounds on the lifting coefficients, which is useful in solving the separation problem of the SL inequalities. The proposed SL inequality is shown to dominate the well-known mixed integer rounding (MIR) inequality under certain conditions. We show that the problem of computing the coefficients for an SL inequality is NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{N}\mathcal{P}$$\end{document}-hard but can be solved by a pseudo-polynomial time algorithm. As a by-product of analysis, we provide new conditions to guarantee the MIR inequality to be facet-defining for the considered polyhedron and prove that in general, the problem of deciding whether an MIR inequality defines a facet is NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{N}\mathcal{P}$$\end{document}-complete. Finally, we perform numerical experiments to evaluate the performance and impact of using the proposed SL inequalities as cutting planes in solving mixed integer linear programming problems. Numerical results demonstrate that the proposed SL cuts are much more effective than the MIR cuts in terms of strengthening the problem formulation and improving the solution efficiency. Moreover, when applied to solve random and real application problems, the proposed SL cuts demonstrate the benefit in reducing the solution time.