Geodesic
A lexicographic $0,5$-approximation algorithm for the multiple knapsack problem
Sibirskij žurnal industrialʹnoj matematiki, Tome 21 (2018) no. 2, pp. 108-121

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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 $A\times B$ (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 $(a,b)\in A\times B$ 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 $m$ and $n$ are the sizes of $A$ and $B$ correspondingly.
Keywords: multiple knapsack problem, lexicographic ordering, approximation algorithm, approximation ratio. 
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     author = {A. B. Khutoretskii and S. V. Bredikhin and A. A. Zamyatin},
     title = {A lexicographic $0,5$-approximation algorithm for the multiple knapsack problem},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {108--121},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2018_21_2_a9/}
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A. B. Khutoretskii; S. V. Bredikhin; A. A. Zamyatin. A lexicographic $0,5$-approximation algorithm for the multiple knapsack problem. Sibirskij žurnal industrialʹnoj matematiki, Tome 21 (2018) no. 2, pp. 108-121. http://geodesic.mathdoc.fr/item/SJIM_2018_21_2_a9/