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A Nearly-Linear Time Algorithm for Submodular Maximization with a Knapsack Constraint
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46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Colloquium on Automata, Languages, and Programming (ICALP) - License:
Creative Commons Attribution 3.0 Unported license
- Publication Date: 2019-07-04
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Acknowledgements
This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing.
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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
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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