Greedy Algorithm for Maximization of Non-submodular Functions Subject to Knapsack Constraint

2019 ◽  
pp. 651-662 ◽  
Author(s):  
Zhenning Zhang ◽  
Bin Liu ◽  
Yishui Wang ◽  
Dachuan Xu ◽  
Dongmei Zhang
Keyword(s):  
Greedy Algorithm ◽  
Submodular Functions ◽  
Knapsack Constraint

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 accelerated continuous greedy algorithm for maximizing strong submodular functions

2013 ◽  
Vol 30 (4) ◽  
pp. 1107-1124
Author(s):  
Zengfu Wang ◽  
Bill Moran ◽  
Xuezhi Wang ◽  
Quan Pan
Keyword(s):  
Greedy Algorithm ◽  
Submodular Functions

scholarly journals Regression under Human Assistance

2020 ◽  
Vol 34 (03) ◽  
pp. 2611-2620
Author(s):  
Abir De ◽  
Paramita Koley ◽  
Niloy Ganguly ◽  
Manuel Gomez-Rodriguez
Keyword(s):  
Machine Learning ◽  
Greedy Algorithm ◽  
Real World ◽  
Medical Diagnosis ◽  
Submodular Functions ◽  
Learning Models ◽  
Real World Data ◽  
Alternative Representation ◽  
The Greedy Algorithm ◽  
Machine Learning Models

Decisions are increasingly taken by both humans and machine learning models. However, machine learning models are currently trained for full automation—they are not aware that some of the decisions may still be taken by humans. In this paper, we take a first step towards the development of machine learning models that are optimized to operate under different automation levels. More specifically, we first introduce the problem of ridge regression under human assistance and show that it is NP-hard. Then, we derive an alternative representation of the corresponding objective function as a difference of nondecreasing submodular functions. Building on this representation, we further show that the objective is nondecreasing and satisfies α-submodularity, a recently introduced notion of approximate submodularity. These properties allow a simple and efficient greedy algorithm to enjoy approximation guarantees at solving the problem. Experiments on synthetic and real-world data from two important applications—medical diagnosis and content moderation—demonstrate that the greedy algorithm beats several competitive baselines.

scholarly journals Streaming Algorithms for Maximizing Monotone Submodular Functions Under a Knapsack Constraint

Algorithmica ◽  
2019 ◽  
Vol 82 (4) ◽  
pp. 1006-1032
Author(s):  
Chien-Chung Huang ◽  
Naonori Kakimura ◽  
Yuichi Yoshida
Keyword(s):  
Submodular Functions ◽  
Streaming Algorithms ◽  
Knapsack Constraint

scholarly journals Maximizing Sequence-Submodular Functions and Its Application to Online Advertising

2021 ◽  
Author(s):  
Saeed Alaei ◽  
Ali Makhdoumi ◽  
Azarakhsh Malekian
Keyword(s):  
Objective Function ◽  
Greedy Algorithm ◽  
Online Advertising ◽  
Optimal Solution ◽  
Allocation Problem ◽  
Submodular Functions ◽  
Query Rewriting ◽  
Nondecreasing Sequence

Motivated by applications in online advertising, we consider a class of maximization problems where the objective is a function of the sequence of actions and the running duration of each action. For these problems, we introduce the concepts of sequence-submodularity and sequence-monotonicity, which extend the notions of submodularity and monotonicity from functions defined over sets to functions defined over sequences. We establish that if the objective function is sequence-submodular and sequence-nondecreasing, then there exists a greedy algorithm that achieves [Formula: see text] of the optimal solution. We apply our algorithm and analysis to two applications in online advertising: online ad allocation and query rewriting. We first show that both problems can be formulated as maximizing nondecreasing sequence-submodular functions. We then apply our framework to these two problems, leading to simple greedy approaches with guaranteed performances. In particular, for the online ad allocation problem, the performance of our algorithm is [Formula: see text], which matches the best known existing performance, and for the query rewriting problem, the performance of our algorithm is [Formula: see text], which improves on the best known existing performance in the literature. This paper was accepted by Chung Piaw Teo, optimization.

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.

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