Online Knapsack Problem Under Concave Functions

In this paper, we address an online knapsack problem under concave function f(x), i.e., an item with size x has its profit f(x). We first obtain a simple lower bound max{q, f’(0)/f(1)}, where q ≈ 1.618, then show that this bound is not tight, and give an improved lower bound.

Finally, we find the online algorithm for linear function can be employed to the concave case, and prove its competitive ratio is f’(0)/f(1/q), then we give a refined online algorithm with a competitive ratio f’(0)/f(1) + 1. And we also give optimal algorithms for some piecewise linear functions.

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Recommended citation: Xin Han, Ning Ma, Kazuhisa Makino, He Chen. (2016). "Online Knapsack Problem Under Concave Functions." Frontiers in Algorithmics.
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