Parallel time–space reduction by unbiased filtering for solving the 0/1-Knapsack problem

Abstract

The 0/1-Knapsack problem (0/1-KP) is one of popular NP-hard problems since its optimal solutions are meaningful to data-science computing in real-world applications (i.e., maximum profit of product planning-and-loading, minimum encryption of reliable information-transmission, etc.). Theoretically, the optimal solutions of the 0/1-KP can be solved by the dynamic programming (DP) but in non-polynomial time, which is not reasonable for big data n and massive limited-capacity C. Practically, the hybrid swarm-optimization could improve the 0/1-KP solutions in polynomial time but best fitness-solutions may not be optimal. Therefore, this paper proposes the parallel time–space reduction for solving the 0/1-KP in polynomial time. Our innovation and contribution focus on achieving 1. the fast processing as the FS (feature sorting) and the optimal solutions close to the DP by our unbiased-filtering and 2. the optimized time-complexity O(n/p log2p + C’/p) by our parallel time–space reduction. In this new approach, we integrate the FS method (for the efficient initial solutions) and the optimal DP (on the remaining critical-objects n’ and knapsack-capacity C’), incorporated with the proposed unbiased-filtering. Finally, performance is evaluated by simulation study on a variety of problem-sizes. In experiment, most of our solutions are optimal as comparable to the optimal solutions of the DP.