Improved Streaming Algorithms for Maximizing Monotone Submodular Functions under a Knapsack Constraint
In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in a streaming setting. In such a setting, elements arrive sequentially and at any point in time, and the algorithm can store only a small fraction of the elements that have arrived s...
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| Veröffentlicht in: | Algorithmica Jg. 83; H. 3; S. 879 - 902 |
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| Abstract | In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in a streaming setting. In such a setting, elements arrive sequentially and at any point in time, and the algorithm can store only a small fraction of the elements that have arrived so far. For the special case that all elements have unit sizes (i.e., the cardinality-constraint case), one can find a
(
0.5
-
ε
)
-approximate solution in
O
(
K
ε
-
1
)
space, where
K
is the knapsack capacity (Badanidiyuru
et al.
KDD 2014). The approximation ratio is recently shown to be optimal (Feldman
et al.
STOC 2020). In this work, we propose a
(
0.4
-
ε
)
-approximation algorithm for the knapsack-constrained problem, using space that is a polynomial of
K
and
ε
. This improves on the previous best ratio of
0.363
-
ε
with space of the same order. Our algorithm is based on a careful combination of various ideas to transform multiple-pass streaming algorithms into a single-pass one. |
|---|---|
| AbstractList | In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in a streaming setting. In such a setting, elements arrive sequentially and at any point in time, and the algorithm can store only a small fraction of the elements that have arrived so far. For the special case that all elements have unit sizes (i.e., the cardinality-constraint case), one can find a
(
0.5
-
ε
)
-approximate solution in
O
(
K
ε
-
1
)
space, where
K
is the knapsack capacity (Badanidiyuru
et al.
KDD 2014). The approximation ratio is recently shown to be optimal (Feldman
et al.
STOC 2020). In this work, we propose a
(
0.4
-
ε
)
-approximation algorithm for the knapsack-constrained problem, using space that is a polynomial of
K
and
ε
. This improves on the previous best ratio of
0.363
-
ε
with space of the same order. Our algorithm is based on a careful combination of various ideas to transform multiple-pass streaming algorithms into a single-pass one. In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in a streaming setting. In such a setting, elements arrive sequentially and at any point in time, and the algorithm can store only a small fraction of the elements that have arrived so far. For the special case that all elements have unit sizes (i.e., the cardinality-constraint case), one can find a (0.5−ε)-approximate solution in O(Kε−1) space, where K is the knapsack capacity (Badanidiyuru et al. KDD 2014). The approximation ratio is recently shown to be optimal (Feldman et al. STOC 2020). In this work, we propose a (0.4−ε)-approximation algorithm for the knapsack-constrained problem, using space that is a polynomial of K and ε. This improves on the previous best ratio of 0.363−ε with space of the same order. Our algorithm is based on a careful combination of various ideas to transform multiple-pass streaming algorithms into a single-pass one. |
| Author | Kakimura, Naonori Huang, Chien-Chung |
| Author_xml | – sequence: 1 givenname: Chien-Chung surname: Huang fullname: Huang, Chien-Chung organization: CNRS, École Normale Supérieure – sequence: 2 givenname: Naonori orcidid: 0000-0002-3918-3479 surname: Kakimura fullname: Kakimura, Naonori email: kakimura@math.keio.ac.jp organization: Keio University |
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| Cites_doi | 10.1007/BFb0121195 10.1137/130920277 10.1137/1.9781611973402.110 10.1137/110839655 10.1145/2627692.2627694 10.1145/3242770 10.1145/3087556.3087585 10.1137/1.9781611974782.78 10.1007/978-3-662-47672-7_26 10.1287/moor.1100.0463 10.1145/2623330.2623637 10.1287/moor.7.3.410 10.1145/3188745.3188752 10.1145/956750.956769 10.1145/2187836.2187888 10.1145/2809814 10.1137/080733991 10.1007/s10107-015-0900-7 10.1007/BF01588971 10.1016/S0167-6377(03)00062-2 10.1609/aaai.v32i1.11529 |
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| Keywords | Streaming algorithm Approximation algorithm Submodular functions |
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| References_xml | – reference: Chan, T.H.H., Jiang, S.H.C., Tang, Z.G., Wu, X.: Online submodular maximization problem with vector packing constraint. In: Annual European Symposium on Algorithms (ESA), pp. 24:1–24:14 (2017) – reference: SviridenkoMA note on maximizing a submodular set function subject to a knapsack constraintOper. Res. Lett.20043214143201710710.1016/S0167-6377(03)00062-2 – reference: FisherMLNemhauserGLWolseyLAAn analysis of approximations for maximizing submodular set functions iMath. Program.19781426529450386610.1007/BF01588971 – reference: LeeJSviridenkoMVondrákJSubmodular maximization over multiple matroids via generalized exchange propertiesMath. Oper. Res.2010354795806277751510.1287/moor.1100.0463 – reference: Chekuri, C., Gupta, S., Quanrud, K.: Streaming algorithms for submodular function maximization. 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| Title | Improved Streaming Algorithms for Maximizing Monotone Submodular Functions under a Knapsack Constraint |
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