Generating Functions in the Knapsack Problem

The knapsack problem with Boolean variables and a single constraint is considered. Combinatorial formulas for calculating and estimating the cardinality of the set of feasible solutions and the values of the functional in various cases depending on given parameters of the problem are derived. The coefficients of the objective function and of the constraint vector and the knapsack size are used as parameters. The baseline technique is the classical method of generating functions. The results obtained can be used to estimate the complexity of enumeration and decomposition methods for solving the problem and can also be used as auxiliary procedures in developing such methods.