Robust monotone submodular function maximization
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10107_2018_1320_ReferencePDF.pdf
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Author(s)
Orlin, James B
Schulz, Andreas S
Udwani, Rajan
Date Issued
September 15, 2018
Publisher
Springer Berlin Heidelberg
Version
Author's final manuscript
Abstract
Abstract
We consider a robust formulation, introduced by Krause et al. (J Artif Intell Res 42:427–486, 2011), of the classical cardinality constrained monotone submodular function maximization problem, and give the first constant factor approximation results. The robustness considered is w.r.t. adversarial removal of up to
$$\tau $$
τ
elements from the chosen set. For the fundamental case of
$$\tau =1$$
τ
=
1
, we give a deterministic
$$(1-1/e)-1/\varTheta (m)$$
(
1
-
1
/
e
)
-
1
/
Θ
(
m
)
approximation algorithm, where m is an input parameter and number of queries scale as
$$O(n^{m+1})$$
O
(
n
m
+
1
)
. In the process, we develop a deterministic
$$(1-1/e)-1/\varTheta (m)$$
(
1
-
1
/
e
)
-
1
/
Θ
(
m
)
approximate greedy algorithm for bi-objective maximization of (two) monotone submodular functions. Generalizing the ideas and using a result from Chekuri et al. (in: FOCS 10, IEEE, pp 575–584, 2010), we show a randomized
$$(1-1/e)-\epsilon $$
(
1
-
1
/
e
)
-
ϵ
approximation for constant
$$\tau $$
τ
and
$$\epsilon \le \frac{1}{\tilde{\varOmega }(\tau )}$$
ϵ
≤
1
Ω
~
(
τ
)
, making
$$O(n^{1/\epsilon ^3})$$
O
(
n
1
/
ϵ
3
)
queries. Further, for
$$\tau \ll \sqrt{k}$$
τ
≪
k
, we give a fast and practical 0.387 algorithm. Finally, we also give a black box result result for the much more general setting of robust maximization subject to an Independence System.
MIT Department
Sloan School of Management
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s10107-018-1320-2
