Efficient 0/1-Multiple-Knapsack Problem Solving by Hybrid DP Transformation and Robust Unbiased Filtering

Abstract

The multiple knapsack problem (0/1-mKP) is a valuable NP-hard problem involved in many science-and-engineering applications. In current research, there exist two main approaches: 1. the exact algorithms for the optimal solutions (i.e., branch-and-bound, dynamic programming (DP), etc.) and 2. the approximate algorithms in polynomial time (i.e., Genetic algorithm, swarm optimization, etc.). In the past, the exact-DP could find the optimal solutions of the 0/1-KP (one knapsack, n objects) in O(nC). For large n and massive C, the unbiased filtering was incorporated with the exact-DP to solve the 0/1-KP in O(n + C′) with 95% optimal solutions. For the complex 0/1-mKP (m knapsacks) in this study, we propose a novel research track with hybrid integration of DP-transformation (DPT), exact-fit (best) knapsack order (m!-to-m2 reduction), and robust unbiased filtering. First, the efficient DPT algorithm is proposed to find the optimal solutions for each knapsack in O([n2,nC]). Next, all knapsacks are fulfilled by the exact-fit (best) knapsack order in O(m2[n2,nC]) over O(m![n2,nC]) while retaining at least 99% optimal solutions as m! orders. Finally, robust unbiased filtering is incorporated to solve the 0/1-mKP in O(m2n). In experiments, our efficient 0/1-mKP reduction confirmed 99% optimal solutions on random and benchmark datasets (n δ 10,000, m δ 100).