An improved binary search algorithm for the Multiple-Choice Knapsack Problem
He, Cheng1 ; Leung, Joseph Y-T.2 ; Lee, Kangbok3 ; 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, Special issue - Advanced Optimization Approaches and Modern OR-Applications, Tome 50 (2016) no. 4-5, pp. 995-1001.

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 + ( 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.

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)
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. 
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     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},
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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, Special issue - Advanced Optimization Approaches and Modern OR-Applications, Tome 50 (2016) no. 4-5, pp. 995-1001. doi : 10.1051/ro/2015061. http://archive.numdam.org/articles/10.1051/ro/2015061/

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