Streaming submodular maximization under d-knapsack constraints

Submodular optimization is a key topic in combinatorial optimization, which has attracted extensive attention in the past few years. Among the known results, most of the submodular functions are defined on set. But recently some progress has been made on the integer lattice. In this paper, we study...

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Published in:	Journal of combinatorial optimization Vol. 45; no. 1; p. 15
Main Authors:	Chen, Zihan, Liu, Bin, Du, Hongmin W.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.01.2023 Springer Nature B.V
Subjects:	Algorithms Approximation Combinatorial analysis Combinatorics Complexity Constraints Convex and Discrete Geometry Design Integers Lower bounds Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Maximization Operations Research/Decision Theory Optimization Sensors Theory of Computation Streaming algorithm Integer lattice Knapsack constraints Noise
ISSN:	1382-6905, 1573-2886
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Abstract	Submodular optimization is a key topic in combinatorial optimization, which has attracted extensive attention in the past few years. Among the known results, most of the submodular functions are defined on set. But recently some progress has been made on the integer lattice. In this paper, we study two problem of maximizing submodular functions with d -knapsack constraints. First, for the problem of maximizing DR-submodular functions with d -knapsack constraints on the integer lattice, we propose a one pass streaming algorithm that achieves a 1 - θ 1 + d -approximation with O log ( d β - 1 ) β ϵ memory complexity and O log ( d β - 1 ) ϵ log ‖ b ‖ ∞ update time per element, where θ = min ( α + ϵ , 0.5 + ϵ ) and α , β are the upper and lower bounds for the cost of each item in the stream. Then we devise an improved streaming algorithm to reduce the memory complexity to O ( d β ϵ ) with unchanged approximation ratio and query complexity. Then for the problem of maximizing submodular functions with d -knapsack constraints under noise, we design a one pass streaming algorithm. When ε → 0 , it achieves a 1 1 - α + d -approximate ratio, memory complexity O log ( d β - 1 ) β ϵ and query complexity O log ( d β - 1 ) ϵ per element. As far as we know, these two are the first streaming algorithms under their corresponding problems.
AbstractList	Submodular optimization is a key topic in combinatorial optimization, which has attracted extensive attention in the past few years. Among the known results, most of the submodular functions are defined on set. But recently some progress has been made on the integer lattice. In this paper, we study two problem of maximizing submodular functions with d-knapsack constraints. First, for the problem of maximizing DR-submodular functions with d-knapsack constraints on the integer lattice, we propose a one pass streaming algorithm that achieves a 1-θ1+d-approximation with Olog(dβ-1)βϵ memory complexity and Olog(dβ-1)ϵlog‖b‖∞ update time per element, where θ=min(α+ϵ,0.5+ϵ) and α,β are the upper and lower bounds for the cost of each item in the stream. Then we devise an improved streaming algorithm to reduce the memory complexity to O(dβϵ) with unchanged approximation ratio and query complexity. Then for the problem of maximizing submodular functions with d-knapsack constraints under noise, we design a one pass streaming algorithm. When ε→0, it achieves a 11-α+d-approximate ratio, memory complexity Olog(dβ-1)βϵ and query complexity Olog(dβ-1)ϵ per element. As far as we know, these two are the first streaming algorithms under their corresponding problems. Submodular optimization is a key topic in combinatorial optimization, which has attracted extensive attention in the past few years. Among the known results, most of the submodular functions are defined on set. But recently some progress has been made on the integer lattice. In this paper, we study two problem of maximizing submodular functions with d -knapsack constraints. First, for the problem of maximizing DR-submodular functions with d -knapsack constraints on the integer lattice, we propose a one pass streaming algorithm that achieves a 1 - θ 1 + d -approximation with O log ( d β - 1 ) β ϵ memory complexity and O log ( d β - 1 ) ϵ log ‖ b ‖ ∞ update time per element, where θ = min ( α + ϵ , 0.5 + ϵ ) and α , β are the upper and lower bounds for the cost of each item in the stream. Then we devise an improved streaming algorithm to reduce the memory complexity to O ( d β ϵ ) with unchanged approximation ratio and query complexity. Then for the problem of maximizing submodular functions with d -knapsack constraints under noise, we design a one pass streaming algorithm. When ε → 0 , it achieves a 1 1 - α + d -approximate ratio, memory complexity O log ( d β - 1 ) β ϵ and query complexity O log ( d β - 1 ) ϵ per element. As far as we know, these two are the first streaming algorithms under their corresponding problems.
ArticleNumber	15
Author	Chen, Zihan Liu, Bin Du, Hongmin W.
Author_xml	– sequence: 1 givenname: Zihan surname: Chen fullname: Chen, Zihan organization: School of Mathematical Sciences, Ocean University of China – sequence: 2 givenname: Bin orcidid: 0000-0002-8958-3999 surname: Liu fullname: Liu, Bin email: binliu@ouc.edu.cn organization: School of Mathematical Sciences, Ocean University of China – sequence: 3 givenname: Hongmin W. surname: Du fullname: Du, Hongmin W. organization: Accounting and Information Systems Department, Rutgers University
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Cites_doi	10.1016/S0167-6377(03)00062-2 10.1007/s00453-020-00786-4 10.1109/ACCESS.2018.2871668 10.1007/s10107-018-1324-y 10.1109/TKDE.2018.2854182 10.1007/BF02680565 10.1109/TNSE.2020.2993042 10.1287/moor.7.3.410 10.1007/BF01588971 10.1109/TCSS.2021.3061452 10.1061/(ASCE)0733-9496(2008)134:6(516) 10.1145/3447396 10.1287/moor.3.3.177 10.1109/ICDCS.2019.00042 10.1145/2623330.2623637 10.1145/956750.956769 10.1137/1.9781611973730.80 10.1145/1835804.1835934 10.1137/1.9781611973105.88 10.1137/1.9781611973068.60 10.1145/1281192.1281239
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SubjectTerms	Algorithms Approximation Combinatorial analysis Combinatorics Complexity Constraints Convex and Discrete Geometry Design Integers Lower bounds Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Maximization Operations Research/Decision Theory Optimization Sensors Theory of Computation
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Title	Streaming submodular maximization under d-knapsack constraints
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