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Subquadratic Submodular Maximization with a General Matroid Constraint

Authors Yusuke Kobayashi , Tatsuya Terao

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Author Details

Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Tatsuya Terao
  • Research Institute for Mathematical Sciences, Kyoto University, Japan

Funding

This work was partially supported by the joint project of Kyoto University and Toyota Motor Corporation, titled "Advanced Mathematical Science for Mobility Society", by JST ERATO Grant Number JPMJER2310, and by JSPS KAKENHI Grant Numbers JP20K11692, JP22H05001, JP24KJ1494, and JP24K02901.

Acknowledgements

The authors thank the three anonymous reviewers for their valuable comments.

Cite AsGet BibTex

Yusuke Kobayashi and Tatsuya Terao. Subquadratic Submodular Maximization with a General Matroid Constraint. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 100:1-100:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.100

Abstract

We consider fast algorithms for monotone submodular maximization with a general matroid constraint. We present a randomized (1 - 1/e - ε)-approximation algorithm that requires Õ_{ε}(√r n) independence oracle and value oracle queries, where n is the number of elements in the matroid and r ≤ n is the rank of the matroid. This improves upon the previously best algorithm by Buchbinder-Feldman-Schwartz [Mathematics of Operations Research 2017] that requires Õ_{ε}(r² + √rn) queries. Our algorithm is based on continuous relaxation, as with other submodular maximization algorithms in the literature. To achieve subquadratic query complexity, we develop a new rounding algorithm, which is our main technical contribution. The rounding algorithm takes as input a point represented as a convex combination of t bases of a matroid and rounds it to an integral solution. Our rounding algorithm requires Õ(r^{3/2} t) independence oracle queries, while the previously best rounding algorithm by Chekuri-Vondrák-Zenklusen [FOCS 2010] requires O(r² t) independence oracle queries. A key idea in our rounding algorithm is to use a directed cycle of arbitrary length in an auxiliary graph, while the algorithm of Chekuri-Vondrák-Zenklusen focused on directed cycles of length two.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithm design techniques
  • Theory of computation → Submodular optimization and polymatroids
Keywords
  • submodular maximization
  • matroid constraint
  • approximation algorithm
  • rounding algorithm
  • query complexity

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