On Maximizing Sums of Non-monotone Submodular and Linear Functions

We study the problem of Regularized Unconstrained SubmodularMaximization (RegularizedUSM) as defined by Bodek and Feldman (Maximizing sums of non-monotone submodular and linear functions: understanding the unconstrained case, arXiv:2204.03412 , 2022): given query access to a non-negative submodular...

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Veröffentlicht in:Algorithmica Jg. 86; H. 4; S. 1080 - 1134
1. Verfasser: Qi, Benjamin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.04.2024
Springer Nature B.V
Schlagworte:
Algorithm Analysis and Problem Complexity
Algorithms
Approximation
Computer Science
Computer Systems Organization and Communication Networks
Constraints
Data Structures and Information Theory
Linear functions
Machine learning
Mathematical analysis
Mathematics of Computing
Maximization
Optimization
Theory of Computation
Double greedy
Continuous greedy
Submodular maximization
Regularization
Inapproximability
ISSN:0178-4617, 1432-0541
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Zusammenfassung:We study the problem of Regularized Unconstrained SubmodularMaximization (RegularizedUSM) as defined by Bodek and Feldman (Maximizing sums of non-monotone submodular and linear functions: understanding the unconstrained case, arXiv:2204.03412 , 2022): given query access to a non-negative submodular function f : 2 N → R ≥ 0 and a linear function ℓ : 2 N → R over the same ground set N , output a set T ⊆ N approximately maximizing the sum f ( T ) + ℓ ( T ) . An algorithm is said to provide an ( α , β ) -approximation for RegularizedUSM if it outputs a set T such that E [ f ( T ) + ℓ ( T ) ] ≥ max S ⊆ N [ α · f ( S ) + β · ℓ ( S ) ] . We also consider the setting where S and T are constrained to be independent in a given matroid, which we refer to as Regularized  Constrained Submodular Maximization (RegularizedCSM). The special case of RegularizedCSM with monotone f has been extensively studied (Sviridenko et al. in Math Oper Res 42(4):1197–1218, 2017; Feldman in Algorithmica 83(3):853–878, 2021; Harshaw et al., in: International conference on machine learning, PMLR, 2634–2643, 2019), whereas we are aware of only one prior work that studies RegularizedCSM with non-monotone f (Lu et al. in Optimization 1–27, 2023), and that work constrains ℓ to be non-positive. In this work, we provide improved ( α , β ) -approximation algorithms for both RegularizedUSM and RegularizedCSM with non-monotone f . Specifically, we are the first to provide nontrivial ( α , β ) -approximations for RegularizedCSM where the sign of ℓ is unconstrained, and the α we obtain for RegularizedUSM improves over (Bodek and Feldman in Maximizing sums of non-monotone submodular and linear functions: understanding the unconstrained case, arXiv:2204.03412 , 2022) for all β ∈ ( 0 , 1 ) . We also prove new inapproximability results for RegularizedUSM and RegularizedCSM, as well as 0.478-inapproximability for maximizing a submodular function where S and T are subject to a cardinality constraint, improving a 0.491-inapproximability result due to Oveis Gharan and Vondrak (in: Proceedings of the twenty-second annual ACM-SIAM symposium on discrete algorithms, SIAM, pp 1098–1116, 2011).
Bibliographie:ObjectType-Article-1
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-023-01183-3