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Streaming algorithms for maximizing the difference of submodular functions and the sum of submodular and supermodular functions

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Abstract

In this paper, we study the problem of maximizing the Difference of two Submodular (DS) functions in the streaming model, where elements in the ground set arrive one at a time in an arbitrary order. We present one-pass streaming algorithms for both the unconstrained and cardinality-constrained problems. Our analysis shows that the algorithms we propose are able to produce solutions with provable approximation guarantees. To the best of our knowledge, this is the first theoretical guarantee for the DS maximization problem in the streaming model. In addition, we study the function maximization problem under a cardinality constraint, where the underlying objective function is a \(\gamma \)-weakly DR-submodular function, in the streaming setting. We propose a one-pass streaming algorithm, which achieves an approximation ratio of \(\gamma /(1 + \gamma ) - \epsilon \). Since the sum of suBmodular and suPermodular (BP) functions can be regarded as a \((1 - \kappa ^g)\)-weakly DR-submodular function, we obtain a \(( (1 - \kappa ^g)/(2 - \kappa ^g) - \epsilon )\)-approximation for the cardinality-constrained BP maximization, where \(\kappa ^g\) is the curvature of the corresponding supermodular function. Our results improve the previous best approximation bounds.

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Notes

  1. Set function f is called monotone if \(f(A) \le f(B)\) for every \(A \subseteq B \subseteq \varOmega \).

  2. For instance, the approximation bound usually takes the form of \(1-e^{-\gamma }\), where \(\gamma \) is the parameter defined to measure the distance between a nonsubmodular function and a submodular one.

  3. A monotone set function \(f :2^\varOmega \rightarrow \mathbb {R}_{\ge 0} \) is called \(\gamma \)-weakly DR-submodular if \(f(e|A) \ge \gamma \cdot f(e|B)\) holds for every \(A \subseteq B \subset \varOmega \) and \(e \in \varOmega \setminus B\).

  4. Throughout this paper, we denote set \(\{1,2,\ldots ,r\}\) by [r] for every positive integer r.

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Acknowledgements

We thank the Associate Editor and two anonymous reviewers for their insightful comments. This research is supported by the National Natural Science Foundation of China under Grant Numbers 11991022 and 12071459 and the Fundamental Research Funds for the Central Universities under Grant Number E1E40107X2.

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  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

    Cheng Lu, Wenguo Yang & Suixiang Gao

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  1. Cheng Lu
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Lu, C., Yang, W. & Gao, S. Streaming algorithms for maximizing the difference of submodular functions and the sum of submodular and supermodular functions. Optim Lett 17, 1643–1667 (2023). https://doi.org/10.1007/s11590-023-01979-w

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