On Maximizing Sums of Non-monotone Submodular and Linear Functions

Abstract

Abstract We study the problem of Regularized Unconstrained Submodular Maximization (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

$$f:2^{{\mathcal {N}}}\rightarrow {\mathbb {R}}_{\ge 0}$$

f :

2 N

→

R

≥ 0

and a linear function

$$\ell :2^{{\mathcal {N}}}\rightarrow {\mathbb {R}}$$

ℓ :

2 N

→ R

over the same ground set

$${\mathcal {N}}$$

N

, output a set

$$T\subseteq {\mathcal {N}}$$

T ⊆ N

approximately maximizing the sum

$$f(T)+\ell (T)$$

f ( T ) + ℓ ( T )

. An algorithm is said to provide an

$$(\alpha ,\beta )$$

( α , β )

-approximation for RegularizedUSM if it outputs a set T such that

$${\mathbb {E}}[f(T)+\ell (T)]\ge \max _{S\subseteq {\mathcal {N}}}[\alpha \cdot f(S)+\beta \cdot \ell (S)]$$

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

$$\ell $$

ℓ

to be non-positive. In this work, we provide improved

$$(\alpha ,\beta )$$

( α , β )

-approximation algorithms for both RegularizedUSM and RegularizedCSM with non-monotone f. Specifically, we are the first to provide nontrivial

$$(\alpha ,\beta )$$

( α , β )

-approximations for RegularizedCSM where the sign of

$$\ell $$

ℓ

is unconstrained, and the

$$\alpha $$

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

$$\beta \in (0,1)$$

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).