On Maximizing Sums of Non-monotone Submodular and Linear Functions - Qi, Benjamin (Algorithmica, 2024)
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).