Abstract
A new technique called balancing is presented for the solution of Knapsack Problems. It is proved that an optimal solution to the Knapsack Problem is balanced, and thus only balanced feasible solutions need to be enumerated in order to solve the problem to optimality. Restricting a dynamic programming algorithm to only consider balanced states implies that the Subset-sum Problem, 0–1 Knapsack Problem, Multiple-choice Subset-sum Problem, and Bounded Knapsack Problem all are solvable in linear time, provided that the weights and profits are bounded by a constant. Extensive computational experiments are presented to document that the derived algorithm for the Subset-sum Problem is able to solve several problems from the literature which could not be solved previously.
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