# A Lexicographic 0.5-Approximation Algorithm for the Multiple Knapsack Problem

A Lexicographic 0.5-Approximation Algorithm for the Multiple Knapsack Problem We present a 0.5-approximation algorithm for the Multiple Knapsack Problem (MKP). The algorithm uses the ordering of knapsacks according to the nondecreasing of size and the two orderings of items: in nonincreasing utility order and in nonincreasing order of the utility/size ratio. These orderings create two lexicographic orderings on A × B (here A is the set of knapsacks and B is the set of indivisible items). Based on each of these lexicographic orderings, the algorithm creates a feasible solution to the MKP by looking through the pairs (a, b) ∈ A × B in the corresponding order and placing item b into knapsack a if this item is not placed yet and there is enough free space in the knapsack. The algorithm chooses the best of the two obtained solutions. This algorithm is 0.5-approximate and has runtime O(mn) (without sorting), where mand n are the sizes of A and B correspondingly. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied and Industrial Mathematics Springer Journals

# A Lexicographic 0.5-Approximation Algorithm for the Multiple Knapsack Problem

, Volume 12 (2) – May 29, 2018
14 pages

/lp/springer_journal/a-lexicographic-0-5-approximation-algorithm-for-the-multiple-knapsack-QiEzFYI9y4
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
1990-4789
eISSN
1990-4797
D.O.I.
10.1134/S1990478918020072
Publisher site
See Article on Publisher Site

### Abstract

We present a 0.5-approximation algorithm for the Multiple Knapsack Problem (MKP). The algorithm uses the ordering of knapsacks according to the nondecreasing of size and the two orderings of items: in nonincreasing utility order and in nonincreasing order of the utility/size ratio. These orderings create two lexicographic orderings on A × B (here A is the set of knapsacks and B is the set of indivisible items). Based on each of these lexicographic orderings, the algorithm creates a feasible solution to the MKP by looking through the pairs (a, b) ∈ A × B in the corresponding order and placing item b into knapsack a if this item is not placed yet and there is enough free space in the knapsack. The algorithm chooses the best of the two obtained solutions. This algorithm is 0.5-approximate and has runtime O(mn) (without sorting), where mand n are the sizes of A and B correspondingly.

### Journal

Journal of Applied and Industrial MathematicsSpringer Journals

Published: May 29, 2018

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