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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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
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ISSN:1382-6905
1573-2886
DOI:10.1007/s10878-022-00951-1