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Weakly Approximating Knapsack in Subquadratic Time

Authors Lin Chen , Jiayi Lian , Yuchen Mao , Guochuan Zhang

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  • Theory of computation → Approximation algorithms analysis
Keywords
  • Knapsack
  • FPTAS

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Abstract

We consider the classic Knapsack problem. Let t and OPT be the capacity and the optimal value, respectively. If one seeks a solution with total profit at least OPT/(1 + ε) and total weight at most t, then Knapsack can be solved in Õ(n + (1/(ε))²) time [Chen, Lian, Mao, and Zhang '24][Mao '24]. This running time is the best possible (up to a logarithmic factor), assuming that (min,+)-convolution cannot be solved in truly subquadratic time [Künnemann, Paturi, and Schneider '17][Cygan, Mucha, Węgrzycki, and Włodarczyk '19]. The same upper and lower bounds hold if one seeks a solution with total profit at least OPT and total weight at most (1 + ε)t. Therefore, it is natural to ask the following question. 
If one seeks a solution with total profit at least OPT/(1+ε) and total weight at most (1 + ε)t, can Knsapck be solved in Õ(n + (1/(ε))^{2-δ}) time for some constant δ > 0? 
We answer this open question affirmatively by proposing an Õ(n + (1/(ε))^{7/4})-time algorithm.

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Lin Chen, Jiayi Lian, Yuchen Mao, and Guochuan Zhang. Weakly Approximating Knapsack in Subquadratic Time. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.51

Author Details

Lin Chen
  • Zhejiang University, Hangzhou, China
Jiayi Lian
  • Zhejiang University, Hangzhou, China
Yuchen Mao
  • Zhejiang University, Hangzhou, China
Guochuan Zhang
  • Zhejiang University, Hangzhou, China

Funding

  • Mao, Yuchen: National Natural Science Foundation of China [Project No. 62402436].
  • Zhang, Guochuan: National Natural Science Foundation of China [Project No. 12271477].

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