Online unit profit knapsack with untrusted predictions

Joan Boyar*, Lene M. Favrholdt, Kim S. Larsen

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review

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Abstract

A variant of the online knapsack problem is considered in the settings of trusted and untrusted predictions. In Unit Profit Knapsack, the items have unit profit, and it is easy to find an optimal solution offline: Pack as many of the smallest items as possible into the knapsack. For Online Unit Profit Knapsack, the competitive ratio is unbounded. In contrast, previous work on online algorithms with untrusted predictions generally studied problems where an online algorithm with a constant competitive ratio is known. The prediction, possibly obtained from a machine learning source, that our algorithm uses is the average size of those smallest items that fit in the knapsack. For the prediction error in this hard online problem, we use the ratio r = a a where a is the actual value for this average size and a is the prediction. The algorithm presented achieves a competitive ratio of 1 2r for r ≥ 1 and r 2 for r ≤ 1. Using an adversary technique, we show that this is optimal in some sense, giving a trade-off in the competitive ratio attainable for different values of r. Note that the result for accurate advice, r = 1, is only 12 , but we show that no deterministic algorithm knowing the value a can achieve a competitive ratio better than e-1 e ≈ 0.6321 and present an algorithm with a matching upper bound. We also show that this latter algorithm attains a competitive ratio of r e-1 e for r ≤ 1 and e-r e for 1 ≤ r < e, and no deterministic algorithm can be better for both r < 1 and 1 < r < e.

Original languageEnglish
Title of host publication18th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2022
EditorsArtur Czumaj, Qin Xin
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Publication date1. Jun 2022
Article number20
ISBN (Electronic)9783959772365
DOIs
Publication statusPublished - 1. Jun 2022
Event18th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2022 - Torshavn, Faroe Islands
Duration: 27. Jun 202229. Jun 2022

Conference

Conference18th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2022
Country/TerritoryFaroe Islands
CityTorshavn
Period27/06/202229/06/2022
SeriesLeibniz International Proceedings in Informatics, LIPIcs
Volume227
ISSN1868-8969

Keywords

  • competitive analysis
  • knapsack problem
  • online algorithms
  • untrusted predictions

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