Abstract
Neighborhood search techniques are often employed to deal with combinatorial optimization problems. Previous works got good results in applying a novel neighborhood search methodology called Multi Improvement (MI). First and best improvement are classical approaches for neighborhood exploration, while the MI has emerged due to the advance of new parallel computing technologies. The MI formalizes the concept of heuristic and exact exploration of independent moves for a given neighborhood structure, however, the advantages of an application of MI face the difficulty to select a great set of independent moves (which can be performed simultaneously). Most of the existing implementations of MI select these moves through heuristic methods, while others have succeeded in implementing exact dynamic programming approaches. In this paper, we propose a formal description for the Maximum Multi Improvement Problem (MMIP), as a theoretical background for the MI. Moreover, we develop three dynamic programming algorithms for solving the MMIP, given a solution tour for a Traveling Salesman Problem and neighborhood operators 2-Opt, 3-Opt, and OrOpt-k. The analysis suggests the rise of a new open topic focused on developing novel efficient neighborhood searches.
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Notes
Neighborhood is a set of solutions (neighbors) that can be reached in one move (2-Opt, 3-Opt, OrOpt, Swap, etc.) on actual solution.
Root Mean Square is helpful measure mainly used in analysis of alternating electrical current. In predictions and estimation, RMS error represents the imperfection of the fit. The fit is more ‘perfect’ when RMS error is closer to zero.
\({R}^2\) or R-squared provides a measure of how reliable is the fit. The value of R-squared is inside the range between 0 and 1. Bests fits returns R-squared more closer to 1.
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Acknowledgements
This work was supported by grants E-26/202.868/2016, E-26/203.272/2017, E-26/202.860/2017, E-26/010.101.249/2018, Rio de Janeiro Research Foundation (FAPERJ) and by grants 303726/2017-2, 303130/2017-2, 313777/2018-7, National Council for Scientific and Technological Development (CNPq).
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Nascimento Silva, J.C., Coelho, I.M., Souza, U.S. et al. Finding the Maximum Multi Improvement on neighborhood exploration. Optim Lett 16, 97–115 (2022). https://doi.org/10.1007/s11590-020-01556-5
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DOI: https://doi.org/10.1007/s11590-020-01556-5