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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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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

Acknowledgements

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

Cite AsGet BibTex

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

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.

Subject Classification

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

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References

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