A simple deterministic algorithm for symmetric submodular maximization subject to a knapsack constraint

2020 ◽  
Vol 163 ◽  
pp. 106010
Author(s):  
Georgios Amanatidis ◽  
Georgios Birmpas ◽  
Evangelos Markakis
Keyword(s):  
Deterministic Algorithm ◽  
Knapsack Constraint ◽  
Submodular Maximization

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 Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint

2021 ◽  
Vol 5 (1) ◽  
pp. 1-22
Author(s):  
Jing Tang ◽  
Xueyan Tang ◽  
Andrew Lim ◽  
Kai Han ◽  
Chongshou Li ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Greedy Algorithm ◽  
Branch And Bound ◽  
Upper Bound ◽  
Optimization Problem ◽  
Approximation Factor ◽  
Real World Application ◽  
Efficiency Of Algorithms ◽  
Knapsack Constraint ◽  
Submodular Maximization

Monotone submodular maximization with a knapsack constraint is NP-hard. Various approximation algorithms have been devised to address this optimization problem. In this paper, we revisit the widely known modified greedy algorithm. First, we show that this algorithm can achieve an approximation factor of 0.405, which significantly improves the known factors of 0.357 given by Wolsey and (1-1/e)/2\approx 0.316 given by Khuller et al. More importantly, our analysis closes a gap in Khuller et al.'s proof for the extensively mentioned approximation factor of (1-1/\sqrte )\approx 0.393 in the literature to clarify a long-standing misconception on this issue. Second, we enhance the modified greedy algorithm to derive a data-dependent upper bound on the optimum. We empirically demonstrate the tightness of our upper bound with a real-world application. The bound enables us to obtain a data-dependent ratio typically much higher than 0.405 between the solution value of the modified greedy algorithm and the optimum. It can also be used to significantly improve the efficiency of algorithms such as branch and bound.

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.

scholarly journals Adaptive Influence Maximization

2021 ◽  
Vol 15 (5) ◽  
pp. 1-23
Author(s):  
Jianxiong Guo ◽  
Weili Wu
Keyword(s):  
Seed Set ◽  
Influence Maximization ◽  
Maximization Problem ◽  
Small Subset ◽  
Influence Spread ◽  
Knapsack Constraint ◽  
Real Scenario ◽  
Submodular Maximization ◽  
Greedy Policy ◽  
Influence Maximization Problem

Influence maximization problem attempts to find a small subset of nodes that makes the expected influence spread maximized, which has been researched intensively before. They all assumed that each user in the seed set we select is activated successfully and then spread the influence. However, in the real scenario, not all users in the seed set are willing to be an influencer. Based on that, we consider each user associated with a probability with which we can activate her as a seed, and we can attempt to activate her many times. In this article, we study the adaptive influence maximization with multiple activations (Adaptive-IMMA) problem, where we select a node in each iteration, observe whether she accepts to be a seed, if yes, wait to observe the influence diffusion process; if no, we can attempt to activate her again with a higher cost or select another node as a seed. We model the multiple activations mathematically and define it on the domain of integer lattice. We propose a new concept, adaptive dr-submodularity, and show our Adaptive-IMMA is the problem that maximizing an adaptive monotone and dr-submodular function under the expected knapsack constraint. Adaptive dr-submodular maximization problem is never covered by any existing studies. Thus, we summarize its properties and study its approximability comprehensively, which is a non-trivial generalization of existing analysis about adaptive submodularity. Besides, to overcome the difficulty to estimate the expected influence spread, we combine our adaptive greedy policy with sampling techniques without losing the approximation ratio but reducing the time complexity. Finally, we conduct experiments on several real datasets to evaluate the effectiveness and efficiency of our proposed policies.

scholarly journals Dynamic Markov Model: Password Guessing Using Probability Adjustment Method

2021 ◽  
Vol 11 (10) ◽  
pp. 4607
Author(s):  
Xiaozhou Guo ◽  
Yi Liu ◽  
Kaijun Tan ◽  
Wenyu Mao ◽  
Min Jin ◽  
...  
Keyword(s):  
Markov Model ◽  
Repetition Rate ◽  
Random Sampling ◽  
High Probability ◽  
Deterministic Algorithm ◽  
High Repetition Rate ◽  
Generation Process ◽  
Dynamic Distribution ◽  
Model Based ◽  
Optimal Adjustment

In password guessing, the Markov model is still widely used due to its simple structure and fast inference speed. However, the Markov model based on random sampling to generate passwords has the problem of a high repetition rate, which leads to a low cover rate. The model based on enumeration has a lower cover rate for high-probability passwords, and it is a deterministic algorithm that always generates the same passwords in the same order, making it vulnerable to attack. We design a dynamic distribution mechanism based on the random sampling method. This mechanism enables the probability distribution of passwords to be dynamically adjusted and tend toward uniform distribution strictly during the generation process. We apply the dynamic distribution mechanism to the Markov model and propose a dynamic Markov model. Through comparative experiments on the RockYou dataset, we set the optimal adjustment degree α. Compared with the Markov model without the dynamic distribution mechanism, the dynamic Markov model reduced the repetition rate from 75.88% to 66.50% and increased the cover rate from 37.65% to 43.49%. In addition, the dynamic Markov model had the highest cover rate for high-probability passwords. Finally, the model avoided the lack of a deterministic algorithm, and when it was run five times, it reached almost the same cover rate as OMEN.

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