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Multiple knapsack-constrained monotone DR-submodular maximization on distributive lattice

— Continuous Greedy Algorithm on Median Complex —

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Abstract

We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Because a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodular maximization problem on a set. We propose a (\(1 - 1/e\))-approximation algorithm for this problem. To achieve this result, we generalize the continuous greedy algorithm to distributive lattices: We choose a median complex as a continuous relaxation of the distributive lattice and define the multilinear extension on it. We show that the median complex admits special curves, named uniform linear motions. The multilinear extension of a DR-submodular function is concave along a positive uniform linear motion, which is a key property used in the continuous greedy algorithm.

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 19K20219. The second author is financially supported by JSPS Research Fellowship Grant Number JP19J22607.

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Authors and Affiliations

  1. RIKEN AIP, Tokyo, Japan

    Takanori Maehara

  2. The University of Tokyo, Tokyo, Japan

    So Nakashima

  3. Kyushu University, Fukuoka, Japan

    Yutaro Yamaguchi

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  1. Takanori Maehara
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  2. So Nakashima
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  3. Yutaro Yamaguchi
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Correspondence to Takanori Maehara.

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Maehara, T., Nakashima, S. & Yamaguchi, Y. Multiple knapsack-constrained monotone DR-submodular maximization on distributive lattice. Math. Program. 194, 85–119 (2022). https://doi.org/10.1007/s10107-021-01620-7

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