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Online Knapsack Problem Under Concave Functions

  • Xin HanEmail author
  • Ning Ma
  • Kazuhisa Makino
  • He Chen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10336)

Abstract

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 \(\max \{q, \frac{f'(0)}{f(1)}\}\), where \(q \approx 1.618\), then show that this bound is not tight, and give an improved lower bound. Finally, we find the online algorithm for linear function [8] can be employed to the concave case, and prove its competitive ratio is \(\frac{f'(0)}{f(1/q)}\), then we give a refined online algorithm with a competitive ratio \(\frac{f'(0)}{f(1)} +1\). And we also give optimal algorithms for some piecewise linear functions.

Notes

Acknowledgment

This research was partially supported by NSFC (11101065), RGC (HKU716412E).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Software SchoolDalian University of TechnologyDalianChina
  2. 2.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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