Maximizing the Sum of a Supermodular Function and a Monotone DR-submodular Function Subject to a Knapsack Constraint on the Integer Lattice

2021 ◽  
pp. 68-75
Author(s):  
Jingjing Tan ◽  
Yicheng Xu ◽  
Dongmei Zhang ◽  
Xiaoqing Zhang
Keyword(s):  
Submodular Function ◽  
Integer Lattice ◽  
Knapsack Constraint ◽  
Supermodular Function

Approximation Algorithms for Submodular Data Summarization with a Knapsack Constraint

2021 ◽  
Vol 5 (1) ◽  
pp. 1-31
Author(s):  
Kai Han ◽  
Shuang Cui ◽  
Tianshuai Zhu ◽  
Enpei Zhang ◽  
Benwei Wu ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Fundamental Problem ◽  
Randomized Algorithm ◽  
Deterministic Algorithm ◽  
Submodular Function ◽  
Approximation Ratio ◽  
Performance Bounds ◽  
Data Summarization ◽  
Submodular Optimization ◽  
Knapsack Constraint

Data summarization, i.e., selecting representative subsets of manageable size out of massive data, is often modeled as a submodular optimization problem. Although there exist extensive algorithms for submodular optimization, many of them incur large computational overheads and hence are not suitable for mining big data. In this work, we consider the fundamental problem of (non-monotone) submodular function maximization with a knapsack constraint, and propose simple yet effective and efficient algorithms for it. Specifically, we propose a deterministic algorithm with approximation ratio 6 and a randomized algorithm with approximation ratio 4, and show that both of them can be accelerated to achieve nearly linear running time at the cost of weakening the approximation ratio by an additive factor of ε. We then consider a more restrictive setting without full access to the whole dataset, and propose streaming algorithms with approximation ratios of 8+ε and 6+ε that make one pass and two passes over the data stream, respectively. As a by-product, we also propose a two-pass streaming algorithm with an approximation ratio of 2+ε when the considered submodular function is monotone. To the best of our knowledge, our algorithms achieve the best performance bounds compared to the state-of-the-art approximation algorithms with efficient implementation for the same problem. Finally, we evaluate our algorithms in two concrete submodular data summarization applications for revenue maximization in social networks and image summarization, and the empirical results show that our algorithms outperform the existing ones in terms of both effectiveness and efficiency.

scholarly journals On maximizing a monotone k-submodular function under a knapsack constraint

2022 ◽  
Vol 50 (1) ◽  
pp. 28-31
Author(s):  
Zhongzheng Tang ◽  
Chenhao Wang ◽  
Hau Chan
Keyword(s):  
Submodular Function ◽  
Knapsack Constraint

Monotone submodular maximization over the bounded integer lattice with cardinality constraints

2019 ◽  
Vol 11 (06) ◽  
pp. 1950075
Author(s):  
Lei Lai ◽  
Qiufen Ni ◽  
Changhong Lu ◽  
Chuanhe Huang ◽  
Weili Wu
Keyword(s):  
Approximation Algorithm ◽  
Greedy Algorithm ◽  
Deterministic Algorithm ◽  
Submodular Function ◽  
Integer Lattice ◽  
Cardinality Constraint ◽  
Cardinality Constraints ◽  
Definition Of ◽  
Deterministic Formula ◽  
Submodular Maximization

We consider the problem of maximizing monotone submodular function over the bounded integer lattice with a cardinality constraint. Function [Formula: see text] is submodular over integer lattice if [Formula: see text], [Formula: see text], where ∨ and ∧ represent elementwise maximum and minimum, respectively. Let [Formula: see text], and [Formula: see text], we study the problem of maximizing submodular function [Formula: see text] with constraints [Formula: see text] and [Formula: see text]. A random greedy [Formula: see text]-approximation algorithm and a deterministic [Formula: see text]-approximation algorithm are proposed in this paper. Both algorithms work in value oracle model. In the random greedy algorithm, we assume the monotone submodular function satisfies diminishing return property, which is not an equivalent definition of submodularity on integer lattice. Additionally, our random greedy algorithm makes [Formula: see text] value oracle queries and deterministic algorithm makes [Formula: see text] value oracle queries.

Streaming algorithm for maximizing a monotone non-submodular function under d-knapsack constraint

2019 ◽  
Vol 14 (5) ◽  
pp. 1235-1248 ◽  
Author(s):  
Yanjun Jiang ◽  
Yishui Wang ◽  
Dachuan Xu ◽  
Ruiqi Yang ◽  
Yong Zhang
Keyword(s):  
Submodular Function ◽  
Streaming Algorithm ◽  
Knapsack Constraint

A 1/2-approximation algorithm for maximizing a non-monotone weak-submodular function on a bounded integer lattice

2020 ◽  
Vol 39 (4) ◽  
pp. 1208-1220 ◽  
Author(s):  
Qingqin Nong ◽  
Jiazhu Fang ◽  
Suning Gong ◽  
Dingzhu Du ◽  
Yan Feng ◽  
...  
Keyword(s):  
Approximation Algorithm ◽  
Submodular Function ◽  
Integer Lattice

Maximizing a monotone non-submodular function under a knapsack constraint

2020 ◽  
Author(s):  
Zhenning Zhang ◽  
Bin Liu ◽  
Yishui Wang ◽  
Dachuan Xu ◽  
Dongmei Zhang
Keyword(s):  
Submodular Function ◽  
Knapsack Constraint
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