On Maximizing Sums of Non-monotone Submodular and Linear Functions
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453_2023_Article_1183.pdf
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1.22 MB
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Adobe PDF
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Author(s)
Qi, Benjamin
Date Issued
November 13, 2023
Publisher
Springer US
Citation
Qi, Benjamin. 2023. "On Maximizing Sums of Non-monotone Submodular and Linear Functions."
Version
Final published version
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).
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00453-023-01183-3
