Towards Nearly-Linear Time Algorithms for Submodular Maximization with a Matroid 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


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. Towards Nearly-Linear Time Algorithms for Submodular Maximization with a Matroid Constraint. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a general matroid constraint with a 1 - 1/e - epsilon approximation that achieves a fast running time provided we have a fast data structure for maintaining an approximately maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the first algorithms for these classes of matroids that achieve a nearly-optimal, 1 - 1/e - epsilon approximation, using a nearly-linear number of function evaluations and arithmetic operations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Submodular optimization and polymatroids
  • submodular maximization
  • matroid constraints
  • fast running times


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