A Competitive Analysis of Online Knapsack Problems with Unit Density

24 Pages Posted: 22 Jul 2019

See all articles by Will Ma

Will Ma

Massachusetts Institute of Technology (MIT)

David Simchi-Levi

Massachusetts Institute of Technology (MIT) - School of Engineering

Jinglong Zhao

Massachusetts Institute of Technology (MIT)

Date Written: July 20, 2019

Abstract

We study an online knapsack problem where the items arrive sequentially and must be either immediately packed into the knapsack or irrevocably discarded. Each item has a different size and the objective is to maximize the total size of items packed. While the competitive ratio of deterministic algorithms for this problem is known to be 0, the competitive ratio of randomized algorithms has, surprisingly, not been considered until now. We derive a random-threshold algorithm which is 0.432-competitive, and show that our threshold distribution is optimal.

We also consider the generalization to multiple knapsacks, where an arriving item has a different size in each knapsack and must be placed in at most one. This is equivalent to the Adwords problem where item truncation is not allowed. We derive a randomized algorithm for this problem which is 0.214-competitive.

Keywords: online knapsack, competitive ratio, online approximation algorithms

Suggested Citation

Ma, Will and Simchi-Levi, David and Zhao, Jinglong, A Competitive Analysis of Online Knapsack Problems with Unit Density (July 20, 2019). Available at SSRN: https://ssrn.com/abstract=3423199 or http://dx.doi.org/10.2139/ssrn.3423199

Will Ma

Massachusetts Institute of Technology (MIT) ( email )

77 Massachusetts Avenue
50 Memorial Drive
Cambridge, MA 02139-4307
United States

David Simchi-Levi

Massachusetts Institute of Technology (MIT) - School of Engineering ( email )

MA
United States

Jinglong Zhao (Contact Author)

Massachusetts Institute of Technology (MIT) ( email )

77 Massachusetts Avenue
50 Memorial Drive
Cambridge, MA 02139-4307
United States

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