| dc.description.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). | en_US |
