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On the PTAS Complexity of Multidimensional Knapsack

Authors Ilan Doron-Arad , Ariel Kulik , Pasin Manurangsi

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • d-dimensional Knapsack
  • Multidimensional Knapsack
  • PTAS
  • CSP

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Abstract

We study the d-dimensional knapsack problem. We are given a set of items, each with a d-dimensional cost vector and a profit, along with a d-dimensional budget vector. The goal is to select a set of items that do not exceed the budget in all dimensions and maximize the total profit. A polynomial-time approximation scheme (PTAS) with running time n^{Θ(d/{ε})} has long been known for this problem, where {ε} is the error parameter and n is the encoding size. Despite decades of active research, the best running time of a PTAS has remained O(n^{⌈ d/{ε} ⌉ - d}). Unfortunately, existing lower bounds only cover the special case with two dimensions d = 2, and do not answer whether there is a n^{o(d/({ε)})}-time PTAS for larger values of d. 
In this work, we show that the running times of the best-known PTAS cannot be improved up to a polylogarithmic factor assuming the Exponential Time Hypothesis (ETH). Our techniques are based on a robust reduction from 2-CSP, which embeds 2-CSP constraints into a desired number of dimensions. Then, using a recent result of [Bafna Karthik and Minzer, STOC'25], we succeed in exhibiting tight trade-off between d and {ε} for all regimes of the parameters assuming d is sufficiently large. Informally, our result also shows that under ETH, for any function f there is no f(d/({ε)}) ⋅ n^{õ(d/({ε)})}-time (1-{ε})-approximation for d-dimensional knapsack, where n is the number of items and õ hides polylogarithmic factors in d/({ε)}.

Cite As Get BibTex

Ilan Doron-Arad, Ariel Kulik, and Pasin Manurangsi. On the PTAS Complexity of Multidimensional Knapsack. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 50:1-50:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.50

Author Details

Ilan Doron-Arad
  • MIT, Cambridge, MA, USA
Ariel Kulik
  • Ben-Gurion University of the Negev, Be'er-Sheva, Israel
Pasin Manurangsi
  • Google Research, Bangkok, Thailand

Acknowledgements

Pasin is grateful to Karthik C. S. for insightful discussions on [Karthik C. S. et al., 2023].

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