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.

Two-stage submodular maximization problem beyond nonnegative and monotone

2021 ◽  
pp. 1-16
Author(s):  
Zhicheng Liu ◽  
Hong Chang ◽  
Ran Ma ◽  
Donglei Du ◽  
Xiaoyan Zhang
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Objective Function ◽  
Modular Function ◽  
Submodular Function ◽  
Maximization Problem ◽  
Cardinality Constraint ◽  
Two Stage ◽  
Time Efficiency ◽  
Submodular Maximization

Abstract We consider a two-stage submodular maximization problem subject to a cardinality constraint and k matroid constraints, where the objective function is the expected difference of a nonnegative monotone submodular function and a nonnegative monotone modular function. We give two bi-factor approximation algorithms for this problem. The first is a deterministic $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right),1} \right)$ -approximation algorithm, and the second is a randomized $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right) - \varepsilon ,1} \right)$ -approximation algorithm with improved time efficiency.

k-Submodular maximization with two kinds of constraints

2020 ◽  
pp. 2150036
Author(s):  
Ganquan Shi ◽  
Shuyang Gu ◽  
Weili Wu
Keyword(s):  
Approximation Algorithm ◽  
Submodular Function ◽  
Approximation Ratio ◽  
Size Difference ◽  
Total Size ◽  
Difference Constraints ◽  
Submodular Maximization ◽  
Individual Size

[Formula: see text]-submodular maximization is a generalization of submodular maximization, which requires us to select [Formula: see text] disjoint subsets instead of one subset. Attracted by practical values and applications, we consider [Formula: see text]-submodular maximization with two kinds of constraints. For total size and individual size difference constraints, we present a [Formula: see text]-approximation algorithm for maximizing a nonnegative k-submodular function, running in time [Formula: see text] at worst. Specially, if [Formula: see text] is multiple of [Formula: see text], the approximation ratio can reduce to [Formula: see text], running in time [Formula: see text] at worst. Besides, this algorithm can be applied to [Formula: see text]-bisubmodular achieving [Formula: see text]-approximation running in time [Formula: see text]. Furthermore, if [Formula: see text] is multiple of 2, the approximation ratio can reduce to [Formula: see text], running in time [Formula: see text] at worst. For individual size constraint, there is a [Formula: see text]-approximation algorithm for maximizing a nonnegative [Formula: see text]-submodular function and an nonnegative [Formula: see text]-bisubmodular function, running in time [Formula: see text] and [Formula: see text] respectively, at worst.

Approximating Robust Parameterized Submodular Function Maximization in Large-Scales

2019 ◽  
Vol 36 (04) ◽  
pp. 1950022 ◽  
Author(s):  
Ruiqi Yang ◽  
Dachuan Xu ◽  
Yanjun Jiang ◽  
Yishui Wang ◽  
Dongmei Zhang
Keyword(s):  
Submodular Function ◽  
The Other ◽  
Selection Problem ◽  
Cardinality Constraint ◽  
Two Phase ◽  
Submodular Function Maximization ◽  
Set Function ◽  
Submodular Maximization

We study a robust parameterized submodular function maximization inspired by [Mitrović, S, I Bogunovic, A Norouzi-Fard and Jakub Tarnawski (2017). Streaming robust submodular maximization: A partitioned thresholding approach. In Proc. NIPS, pp. 4560–4569] and [Bogunovic, I, J Zhao and V Cevher (2018). Robust maximization of nonsubmodular objectives. In Proc. AISTATS, pp. 890–899]. In our setting, given a parameterized set function, there are two additional twists. One is that elements arrive in a streaming style, and the other is that there are at most [Formula: see text] items deleted from the algorithm’s memory when the stream is finished. The goal is to choose a robust set from the stream such that the robust ratio is maximized. We propose a two-phase algorithm for maximizing a normalized monotone robust parameterized submodular function with a cardinality constraint and show the robust ratio is close to a constant as [Formula: see text]. In the end, we empirically demonstrate the performance of our algorithm on deletion robust support selection problem.

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

A fast double greedy algorithm for non-monotone DR-submodular function maximization

2019 ◽  
Vol 12 (01) ◽  
pp. 2050007 ◽  
Author(s):  
Shuyang Gu ◽  
Ganquan Shi ◽  
Weili Wu ◽  
Changhong Lu
Keyword(s):  
Machine Learning ◽  
Greedy Algorithm ◽  
Submodular Function ◽  
Integer Lattice ◽  
Submodular Functions ◽  
Running Time ◽  
Submodular Function Maximization ◽  
Multiple Copies ◽  
Submodular Set Function ◽  
Set Function

We study the problem of maximizing non-monotone diminish return (DR)-submodular function on the bounded integer lattice, which is a generalization of submodular set function. DR-submodular functions consider the case that we can choose multiple copies for each element in the ground set. This generalization has many applications in machine learning. In this paper, we propose a [Formula: see text]-approximation algorithm with a running time of [Formula: see text], where [Formula: see text] is the size of the ground set, [Formula: see text] is the upper bound of integer lattice. Discovering important properties of DR-submodular function, we propose a fast double greedy algorithm which improves the running time.

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.

An Attribute Mapping Technique for Secure Interoperation in Multi-Domain Environments

2014 ◽  
Vol 519-520 ◽  
pp. 181-184
Author(s):  
Jian Feng Lu ◽  
Xuan Yan ◽  
Yi Ding Liu
Keyword(s):  
Access Control ◽  
Mapping Technique ◽  
Cardinality Constraint ◽  
Constraint Violation ◽  
Basic Technique ◽  
Access Control Policies ◽  
Role Mapping ◽  
Simple Action ◽  
Definition Of ◽  
Secure Interoperation

Role mapping is a basic technique for facilitating interoperation in RBAC-based collaborating environments. However, role mapping lacks the flexibility to specify access control policies in the scenarios where the access control is not a simple action, but consists of a sequence of actions and events from subjects and system. In this paper, we propose an attribute mapping technique to establish secure context in multi-domain environments. We first classify attributes into eight types and show that only two types of attributes need to be translated. We second give the definition of attribute mapping technique, and analysis the properties of attribute mapping. Finally, we study how cardinality constraint violation arises and shows that it is efficient to resolve this security violation.

Preserving Relationship Cardinality Constraints in Relational Schemata

2002 ◽  
pp. 66-112
Author(s):  
Dolores Cuadra ◽  
Carlos Nieto ◽  
Paloma Martinez ◽  
Elena Castro ◽  
Manuel Velasco
Keyword(s):  
Relational Model ◽  
Future Research ◽  
Methodological Framework ◽  
Cardinality Constraint ◽  
Cardinality Constraints ◽  
Active Rules ◽  
Model Following ◽  
Methodological Approaches ◽  
The Relationship ◽  
Practical Implications

This chapter is devoted to the study of the transformation of conceptual into logical schemata in a methodological framework focusing on a special ER construct: the relationship and its associated cardinality constraints. The section entitled “EER Model Revised: relationships and cardinality constraint” reviews the relationship and cardinality constraint constructs through different methodological approaches to establish the cardinality constraint definition that will be followed in next sections. The section “Transformation of EER Schemata into Relational Schemata” is related to the transformation of conceptual n-ary relationships (n³2) into the relational model following an active rules approach. Finally, several practical implications as well as future research paths are presented.

scholarly journals ON THE PIPAGE ROUNDING ALGORITHM FOR SUBMODULAR FUNCTION MAXIMIZATION — A VIEW FROM DISCRETE CONVEX ANALYSIS

2009 ◽  
Vol 01 (01) ◽  
pp. 1-23 ◽  
Author(s):  
AKIYOSHI SHIOURA
Keyword(s):  
Approximation Algorithm ◽  
Convex Analysis ◽  
Submodular Function ◽  
Submodular Functions ◽  
Concave Functions ◽  
Discrete Convex Analysis ◽  
Discrete Convex ◽  
Weighted Rank ◽  
Rounding Algorithm ◽  
Rank Functions

We consider the problem of maximizing a nondecreasing submodular set function under a matroid constraint. Recently, Calinescu et al. (2007) proposed an elegant framework for the approximation of this problem, which is based on the pipage rounding technique by Ageev and Sviridenko (2004), and showed that this framework indeed yields a (1 - 1/e)-approximation algorithm for the class of submodular functions which are represented as the sum of weighted rank functions of matroids. This paper sheds a new light on this result from the viewpoint of discrete convex analysis by extending it to the class of submodular functions which are the sum of M ♮-concave functions. M ♮-concave functions are a class of discrete concave functions introduced by Murota and Shioura (1999), and contain the class of the sum of weighted rank functions as a proper subclass. Our result provides a better understanding for why the pipage rounding algorithm works for the sum of weighted rank functions. Based on the new observation, we further extend the approximation algorithm to the maximization of a nondecreasing submodular function over an integral polymatroid. This extension has an application in multi-unit combinatorial auctions.

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