Approximation schemes for r-weighted Minimization Knapsack problems

Stimulated by salient applications arising from power systems, this paper studies a class of non-linear Knapsack problems with non-separable quadratic constrains, formulated in either binary or integer form. These problems resemble the duals of the corresponding variants of 2-weighted Knapsack problem (a.k.a., complex-demand Knapsack problem) which has been studied in the extant literature under the paradigm of smart grids. Nevertheless, the employed techniques resulting in a polynomial-time approximation scheme (PTAS) for the 2-weighted Knapsack problem are not amenable to its minimization version. We instead propose a greedy geometry-based approach that arrives at a quasi PTAS (QPTAS) for the minimization variant with boolean variables. As for the integer formulation, a linear programming-based method is developed that obtains a PTAS. In view of the curse of dimensionality, fast greedy heuristic algorithms are presented, additionally to QPTAS. Their performance is corroborated extensively by empirical simulations under diverse settings and scenarios.