A unified approach to polynomially solvable cases of integer “non-separable” quadratic optimization Original Research Article

A recent paper by Hochbaum and Shanthikumar presented “a general-purpose algorithm for converting procedures that solve linear programming problems with … integer variables, to procedures for solving … separable [non-linear] problems”. Their work showed that “convex separable optimization is not much harder than linear optimization”. In contrast, polynomial algorithms in the literature for “non-separable” integer quadratic problems use qualitatively different techniques. By linearly transforming these problems so that the objective is separable in the transformed reference frame, we provide alternative algorithms for these problems based on Hochbaum and Shanthikumarʹs algorithms. Inter alia we introduce a new class of polynomially solvable integer quadratic optimization problems. We also show that a slight generalization of integer linear programming having a non-separable, non-linear objective and totally unimodular constraints in NP-hard.