## 4 0/1-mKP Solving by Multi-DPT-List\({}_{\text{TSR}}\) Plus Unbiased Filtering in Efficient Time
### Efficient Multi-DPT-List\({}_{\text{TSR}}\) Algorithm for Solving 0/1-mKP
#### 4.1.2 The Exact-Fit (Best) Knapsack Order for Regular and Irregular Datasets
if the high optimal performance is the most important factor (in the critical datasets and in the critical 0/1-mKP applications), the integration (of the original multi-DPT-List\({}_{\text{TSR}}\) and the fast multi-DPT-List\({}_{\text{TSR}}\) + robust filtering) provides higher precision (i.e., 99.9% optimal solutions) in efficient O(\(m[n^{2},\,nC]\)) in parallel (\(p=m\)) or O(\(mn^{2}\)) in the best case and O(\(mn^{3}\)) in average if \(C\leq n^{2}\). ## 7 Conclusions

In this study, to solve the complex 0/1-mKP (\(m\) knapsacks) in polynomial time we introduced a novel research track with hybrid integration of DP transformation (for the optimal solution in each knapsack) and robust unbiased filtering (for polynomial time). First, the efficient DPT-List\({}_{\text{TSR}}\) algorithm was proposed to find the optimal solutions of the 0/1-KP in O(\([n^{2},\,nC]\)) over O(\(nC\)) before being applied in the 0/1-mKP. Second, for solving the 0/1-mKP we proposed the multi-DPT-List\({}_{\text{TSR}}\) with the exact-fit (best) knapsack order (\(m!\)-to-\(m^{2}\) reduction) with 99% optimal solutions in O(\(m^{2}[n^{2},\,nC]\)) over O(\(m![n^{2},\,nC]\)). Third (for large \(n\), massive \(C\)), robust unbiased filtering was incorporated into our multi-DPT-List\({}_{\text{TSR}}\) to solve the 0/1-mKP in efficient O(\(m^{2}n\)) over O(\(mnC\)) of the recent HyMKP, while retaining 99% optimal solutions. The experiment was conducted to evaluate the performance of our multi-DPT-List + robust unbiased filtering (with 99% optimal solutions) on random and benchmark datasets (\(n\leq\) 10,000, \(m\leq 100\)). Practically (for large \(m\), \(n\), and \(C\)), our multi-DPT-List\({}_{\text{TSR}}\) + robust unbiased filtering (O(\(m^{2}n\))) could find 99% optimal solutions (as the original multi-DPT-List\({}_{\text{TSR}}\) (O(\(m^{2}[n^{2},\,nC]\))) in polynomial time. In our current research, we apply our multi-DPT-List\({}_{\text{TSR}}\) + robust unbiased filtering to solve the multi-container packing. In the future study, we will modify our unbiased filtering idea to solve another popular NP-hard problem (i.e., traveling salesman and logistic transportation, etc.) in efficient time with expected high optimal performance. Conceptualization, P.B. and J.W. ; methodology, P.B. ; software, P.B. ; validation, P.B. and J.W. ; formal analysis, J.W. ; investigation, P.B. ; writing--original draft preparation, P.B. ; writing--review and editing, J.W. ; visualization, P.B. ; supervision, J.W. All authors have read and agreed to the published version of the manuscript. This research was funded by the School of Science, King Mongkut's Institute of Technology Ladkrabang (KMITL), Bangkok, Thailand, grant number [CW-011-2/2562]. Not applicable. The authors would like to deeply thank the School of Science, King Mongkut's Institute of Technology Ladkrabang (KMITL), Bangkok, Thailand for the Ph.D. research grant (CW-011-2/2562) to [PERSON]. In addition, they would like to gratefully thank [PERSON] ([PERSON]) with his \"Put-ta-tum\" Buddhism-handbook for fulfilling their exact understanding and good decision-making. Last, they would like to gloriously honor their beloved King [PERSON] for the great inspiration in the optimistic study through positive thinking plus critical rethinking. The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results. ## References

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