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A Nearly-Linear Time Algorithm for Submodular Maximization with a Knapsack Constraint

Authors Alina Ene, Huy L. Nguyen

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Subject Classification

ACM Subject Classification
  • Theory of computation → Submodular optimization and polymatroids
Keywords
  • submodular maximization
  • knapsack constraint
  • fast algorithms

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Abstract

We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint. Our main contribution is an algorithm that achieves a nearly-optimal, 1 - 1/e - epsilon approximation, using (1/epsilon)^{O(1/epsilon^4)} n log^2{n} function evaluations and arithmetic operations. Our algorithm is impractical but theoretically interesting, since it overcomes a fundamental running time bottleneck of the multilinear extension relaxation framework. This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to Omega(n^2) running times, since evaluating the multilinear extension is expensive. Our algorithm maintains a fractional solution with only a constant number of entries that are strictly fractional, which allows us to overcome this obstacle.

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Alina Ene and Huy L. Nguyen. A Nearly-Linear Time Algorithm for Submodular Maximization with a Knapsack Constraint. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 53:1-53:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.53

Author Details

Alina Ene
  • Department of Computer Science, Boston University, MA, USA
Huy L. Nguyen
  • College of Computer and Information Science, Northeastern University, Boston, MA, USA

Funding

  • Ene, Alina: Partially supported by NSF CAREER grant CCF-1750333 and NSF grant CCF-1718342.
  • Nguyen, Huy L.: Partially supported by NSF CAREER grant CCF-1750716.

Acknowledgements

This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing.

References

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