scholarly journals Faster Guarantees of Evolutionary Algorithms for Maximization of Monotone Submodular Functions

2021 ◽  
Author(s):  
Victoria G. Crawford
Keyword(s):  
Optimization Algorithm ◽  
Empirical Evaluation ◽  
Pareto Optimization ◽  
Submodular Functions ◽  
Maximization Problem ◽  
Cardinality Constraint ◽  
Worst Case ◽  
Worst Case Ratio ◽  
Submodular Maximization ◽  
Greedy Solution

In this paper, the monotone submodular maximization problem (SM) is studied. SM is to find a subset of size kappa from a universe of size n that maximizes a monotone submodular objective function f . We show using a novel analysis that the Pareto optimization algorithm achieves a worst-case ratio of (1 − epsilon)(1 − 1/e) in expectation for every cardinality constraint kappa < P , where P ≤ n + 1 is an input, in O(nP ln(1/epsilon)) queries of f . In addition, a novel evolutionary algorithm called the biased Pareto optimization algorithm, is proposed that achieves a worst-case ratio of (1 − epsilon)(1 − 1/e − epsilon) in expectation for every cardinality constraint kappa < P in O(n ln(P ) ln(1/epsilon)) queries of f . Further, the biased Pareto optimization algorithm can be modified in order to achieve a a worst-case ratio of (1 − epsilon)(1 − 1/e − epsilon) in expectation for cardinality constraint kappa in O(n ln(1/epsilon)) queries of f . An empirical evaluation corroborates our theoretical analysis of the algorithms, as the algorithms exceed the stochastic greedy solution value at roughly when one would expect based upon our analysis.

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.

Unconstrained submodular maximization with modular costs

2021 ◽  
Vol 14 (10) ◽  
pp. 1756-1768
Author(s):  
Tianyuan Jin ◽  
Yu Yang ◽  
Renchi Yang ◽  
Jieming Shi ◽  
Keke Huang ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Profit Maximization ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Viral Marketing ◽  
Trivial Extension ◽  
Maximization Problem ◽  
Solution Quality ◽  
Worst Case ◽  
Submodular Maximization

Given a set V , the problem of unconstrained submodular maximization with modular costs (USM-MC) asks for a subset S ⊆ V that maximizes f ( S ) - c ( S ), where f is a non-negative, monotone, and submodular function that gauges the utility of S , and c is a non-negative and modular function that measures the cost of S. This problem finds applications in numerous practical scenarios, such as profit maximization in viral marketing on social media. This paper presents ROI-Greedy, a polynomial time algorithm for USM-MC that returns a solution S satisfying [EQUATION], where S * is the optimal solution to USM-MC. To our knowledge, ROI-Greedy is the first algorithm that provides such a strong approximation guarantee. In addition, we show that this worst-case guarantee is tight , in the sense that no polynomial time algorithm can ensure [EQUATION], for any ϵ > 0. Further, we devise a non-trivial extension of ROI-Greedy to solve the profit maximization problem, where the precise value of f ( S ) for any set S is unknown and can only be approximated via sampling. Extensive experiments on benchmark datasets demonstrate that ROI-Greedy significantly outperforms competing methods in terms of the tradeoff between efficiency and solution quality.

scholarly journals Competitive Caching with Machine Learned Advice

2021 ◽  
Vol 68 (4) ◽  
pp. 1-25
Author(s):  
Thodoris Lykouris ◽  
Sergei Vassilvitskii
Keyword(s):  
Online Algorithms ◽  
Empirical Evaluation ◽  
Optimal Solution ◽  
Poor Performance ◽  
Machine Learning Algorithms ◽  
Average Error ◽  
Generalization Error ◽  
Worst Case ◽  
Future Events ◽  
Real World Datasets

Traditional online algorithms encapsulate decision making under uncertainty, and give ways to hedge against all possible future events, while guaranteeing a nearly optimal solution, as compared to an offline optimum. On the other hand, machine learning algorithms are in the business of extrapolating patterns found in the data to predict the future, and usually come with strong guarantees on the expected generalization error. In this work, we develop a framework for augmenting online algorithms with a machine learned predictor to achieve competitive ratios that provably improve upon unconditional worst-case lower bounds when the predictor has low error. Our approach treats the predictor as a complete black box and is not dependent on its inner workings or the exact distribution of its errors. We apply this framework to the traditional caching problem—creating an eviction strategy for a cache of size k . We demonstrate that naively following the oracle’s recommendations may lead to very poor performance, even when the average error is quite low. Instead, we show how to modify the Marker algorithm to take into account the predictions and prove that this combined approach achieves a competitive ratio that both (i) decreases as the predictor’s error decreases and (ii) is always capped by O (log k ), which can be achieved without any assistance from the predictor. We complement our results with an empirical evaluation of our algorithm on real-world datasets and show that it performs well empirically even when using simple off-the-shelf predictions.

scholarly journals Non-submodular model for group profit maximization problem in social networks

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Jianming Zhu ◽  
Smita Ghosh ◽  
Weili Wu ◽  
Chuangen Gao
Keyword(s):  
Social Networks ◽  
Profit Maximization ◽  
Randomized Algorithm ◽  
Political Orientation ◽  
Submodular Functions ◽  
Maximization Problem ◽  
Modular Algorithm ◽  
The Difference ◽  
Group Enterprise ◽  
Theoretical Results

AbstractIn social networks, there exist many kinds of groups in which people may have the same interests, hobbies, or political orientation. Sometimes, group decisions are made by simply majority, which means that most of the users in this group reach an agreement, such as US Presidential Elections. A group is called activated if $$\beta$$ β percent of users are influenced in the group. Enterprise will gain income from all influenced groups. Simultaneously, to propagate influence, enterprise needs pay advertisement diffusion cost. Group profit maximization (GPM) problem aims to pick k seeds to maximize the expected profit that considers the benefit of influenced groups with the diffusion cost. GPM is proved to be NP-hard and the objective function is proved to be neither submodular nor supermodular. An upper bound and a lower bound which are difference of two submodular functions are designed. We propose a submodular–modular algorithm (SMA) to solve the difference of two submodular functions and SMA is shown to converge to a local optimal. We present an randomized algorithm based on weighted group coverage maximization for GPM and apply sandwich framework to get theoretical results. Our experiments verify the efficiency of our methods.

Nonsubmodular Constrained Profit Maximization in Attribute Networks

2021 ◽  
Author(s):  
Liman Du ◽  
Wenguo Yang ◽  
Suixiang Gao
Keyword(s):  
Profit Maximization ◽  
Logistic Function ◽  
Influence Maximization ◽  
Maximization Problem ◽  
Cardinality Constraint ◽  
Three Phase ◽  
Marginal Increment ◽  
The Difference ◽  
Classical Influence ◽  
Influence Maximization Problem

The number of social individuals who interact with their friends through social networks is increasing, leading to an undeniable fact that word-of-mouth marketing has become one of the useful ways to promote sale of products. The Constrained Profit Maximization in Attribute network (CPMA) problem, as an extension of the classical influence maximization problem, is the main focus of this paper. We propose the profit maximization in attribute network problem under a cardinality constraint which is closer to the actual situation. The profit spread metric of CPMA calculates the total benefit and cost generated by all the active nodes. Different from the classical Influence Maximization problem, the influence strength should be recalculated according to the emotional tendency and classification label of nodes in attribute networks. The profit spread metric is no longer monotone and submodular in general. Given that the profit spread metric can be expressed as the difference between two submodular functions and admits a DS decomposition, a three-phase algorithm named as Marginal increment and Community-based Prune and Search(MCPS) Algorithm frame is proposed which is based on Louvain algorithm and logistic function. Due to the method of marginal increment, MPCS algorithm can compute profit spread more directly and accurately. Experiments demonstrate the effectiveness of MCPS algorithm.

scholarly journals A personalized FEM model for reproducible measurement of anti-inflammatory drugs in transdermal administration to knee

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Pasquale Arpaia ◽  
Federica Crauso ◽  
Mirco Frosolone ◽  
Massimo Mariconda ◽  
Simone Minucci ◽  
...  
Keyword(s):  
Optimization Algorithm ◽  
Reconstruction Error ◽  
Average Error ◽  
Element Analysis ◽  
Transdermal Administration ◽  
Worst Case ◽  
Human Knee ◽  
Inflammatory Drugs ◽  
Anti Inflammatory

AbstractA personalized model of the human knee for enhancing the inter-individual reproducibility of a measurement method for monitoring Non-Steroidal Anti-Inflammatory Drugs (NSAIDs) after transdermal delivery is proposed. The model is based on the solution of Maxwell Equations in the electric-quasi-stationary limit via Finite Element Analysis. The dimensions of the custom geometry are estimated on the basis of knee circumference at the patella, body mass index, and sex of each individual. An optimization algorithm allows to find out the electrical parameters of each subject by experimental impedance spectroscopy data. Muscular tissues were characterized anisotropically, by extracting Cole–Cole equation parameters from experimental data acquired with twofold excitation, both transversal and parallel to tissue fibers. A sensitivity and optimization analysis aiming at reducing computational burden in model customization achieved a worst-case reconstruction error lower than 5%. The personalized knee model and the optimization algorithm were validated in vivo by an experimental campaign on thirty volunteers, 67% healthy and 33% affected by knee osteoarthritis (Kellgren–Lawrence grade ranging in [1,4]), with an average error of 3%.

scholarly journals Robust Secure Beamforming Design for Cooperative Cognitive Radio Nonorthogonal Multiple Access Networks

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Quanzhong Li ◽  
Sai Zhao
Keyword(s):  
Cognitive Radio ◽  
Secure Communication ◽  
Multiple Access ◽  
Spectrum Efficiency ◽  
Maximization Problem ◽  
Artificial Noise ◽  
Practical Case ◽  
Secrecy Rate ◽  
Worst Case ◽  
Rank One

By the integration of cooperative cognitive radio (CR) and nonorthogonal multiple access (NOMA), cooperative CR NOMA networks can improve the spectrum efficiency of wireless networks significantly. Due to the openness and exposure of wireless signals, secure communication is an important issue for cooperative CR NOMA networks. In this paper, we investigate the physical layer security design for cooperative CR NOMA networks. Our objective is to achieve maximum secrecy rate of the secondary user by designing optimal beamformers and artificial noise covariance matrix at the multiantenna secondary transmitter under the quality-of-service at the primary user and the transmit power constraint at the secondary transmitter. We consider the practical case that the channel state information (CSI) of the eavesdropper is imperfect, and we model the imperfect CSI by the worst-case model. We show that the robust secrecy rate maximization problem can be transformed to a series of semidefinite programmings based on S-procedure and rank-one relaxation. We also propose an effective method to recover the optimal rank-one solution. Simulations are provided to show the effectiveness of our proposed robust secure algorithm with comparison to the nonrobust secure design and traditional orthogonal multiple access schemes.

An Improved Robust Optimization Algorithm: Second-Order Sensitivity Assisted Worst Case Optimization

2013 ◽  
Vol 49 (5) ◽  
pp. 2109-2112 ◽  
Author(s):  
Ziyan Ren ◽  
Dianhai Zhang ◽  
Chang-Seop Koh
Keyword(s):  
Robust Optimization ◽  
Optimization Algorithm ◽  
Second Order ◽  
Worst Case
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