Nonsubmodular Maximization with Knapsack Constraint via Multilinear Extension

2021 ◽  
pp. 81-92
Author(s):  
Jiachen Ju ◽  
Min Li ◽  
Jianxin Liu ◽  
Qian Liu ◽  
Yang Zhou
Keyword(s):  
Multilinear Extension ◽  
Knapsack Constraint

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.

Fuzzy Bi-cooperative games in multilinear extension form

2015 ◽  
Vol 259 ◽  
pp. 44-55 ◽  
Author(s):  
Surajit Borkotokey ◽  
Pankaj Hazarika ◽  
Radko Mesiar
Keyword(s):  
Cooperative Games ◽  
Multilinear Extension

scholarly journals On using an automatic scheme for obtaining the convex hull defining inequalities of a Weismantel 0–1 knapsack constraint

Top ◽  
1999 ◽  
Vol 7 (1) ◽  
pp. 145-153 ◽  
Author(s):  
Laureano F. Escudero ◽  
Araceli Garín ◽  
Gloria Pérez
Keyword(s):  
Convex Hull ◽  
Knapsack Constraint

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

scholarly journals Bounds for Owen's Multilinear Extension

2007 ◽  
Vol 44 (4) ◽  
pp. 852-864 ◽  
Author(s):  
Josep Freixas
Keyword(s):  
Game Theory ◽  
Simple Game ◽  
Practical Interest ◽  
Simple Games ◽  
Theoretical Interest ◽  
Complex Component ◽  
Multilinear Extension ◽  
Voting System ◽  
Minimal Winning Coalitions ◽  
Minimal Blocking

Owen's multilinear extension (MLE) of a game is a very important tool in game theory and particularly in the field of simple games. Among other applications it serves to efficiently compute several solution concepts. In this paper we provide bounds for the MLE. Apart from its self-contained theoretical interest, the bounds offer the means in voting system studies of approximating the probability that a proposal is approved in a particular simple game having a complex component arrangement. The practical interest of the bounds is that they can be useful for simple games having a tedious MLE to evaluate exactly, but whose minimal winning coalitions and minimal blocking coalitions can be determined by inspection. Such simple games are quite numerous.

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
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