An improved binary search algorithm for the Multiple-Choice Knapsack Problem

He, Cheng; Leung, Joseph Y-T.; Lee, Kangbok; Pinedo, Michael L.

doi:10.1051/ro/2015061

He, Cheng ¹ ; Leung, Joseph Y-T.² ; Lee, Kangbok ³ ; Pinedo, Michael L.⁴

¹ School of Science, Henan University of Technology, Zhengzhou, Henan 450001, P.R. China.
² Department of Computer Science, New Jersey Institute of Technology, Newark NJ-07102, USA.
³ Department of Business and Economics, York College, The City University of New York, 94-20 Guy R. Brewer Blvd, Jamaica, New York 11451, USA.
⁴ Department of Information, Operations and Management Sciences, Stern School of Business, New York University, 44 West 4th Street, New York 10012-1126, USA.

RAIRO - Operations Research - Recherche Opérationnelle, Tome 50 (2016) no. 4-5, pp. 995-1001.

Résumé

The Multiple-Choice Knapsack Problem is defined as a 0-1 Knapsack Problem with additional disjoint multiple-choice constraints. Gens and Levner presented for this problem an approximate binary search algorithm with a worst case ratio of $5$ . We present an improved approximate binary search algorithm with a ratio of $3 + {(\frac{1}{2})}^{t}$ and a running time $O (n (t + log m))$ , where $n$ is the number of items, $m$ the number of classes, and $t$ a positive integer. We then extend our algorithm to make it also applicable to the Multiple-Choice Multidimensional Knapsack Problem with dimension $d$ .

MR Zbl

DOI : 10.1051/ro/2015061

Classification : 68Q25, 90C10, 90C27
Mots clés : Multiple-Choice Knapsack Problem (MCKP), Approximate binary search algorithm, Worst-case performance ratio, Multiple-choice Multi-dimensional Knapsack Problem (MMKP)

Affiliations des auteurs :

He, Cheng ¹ ; Leung, Joseph Y-T. ² ; Lee, Kangbok ³ ; Pinedo, Michael L. ⁴

@article{RO_2016__50_4-5_995_0,
     author = {He, Cheng and Leung, Joseph Y-T. and Lee, Kangbok and Pinedo, Michael L.},
     title = {An improved binary search algorithm for the {Multiple-Choice} {Knapsack} {Problem}},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {995--1001},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {4-5},
     year = {2016},
     doi = {10.1051/ro/2015061},
     mrnumber = {3570544},
     zbl = {1401.90191},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro/2015061/}
}

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He, Cheng; Leung, Joseph Y-T.; Lee, Kangbok; Pinedo, Michael L. An improved binary search algorithm for the Multiple-Choice Knapsack Problem. RAIRO - Operations Research - Recherche Opérationnelle, Tome 50 (2016) no. 4-5, pp. 995-1001. doi : 10.1051/ro/2015061. http://www.numdam.org/articles/10.1051/ro/2015061/

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