scholarly journals Improved Randomized Algorithm for k-Submodular Function Maximization

2021 ◽  
Vol 35 (1) ◽  
pp. 1-22
Author(s):  
Hiroki Oshima
Keyword(s):  
Randomized Algorithm ◽  
Submodular Function ◽  
Submodular Function 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 Very Fast Streaming Submodular Function Maximization

2021 ◽  
pp. 151-166
Author(s):  
Sebastian Buschjäger ◽  
Philipp-Jan Honysz ◽  
Lukas Pfahler ◽  
Katharina Morik
Keyword(s):  
Submodular Function ◽  
Submodular Function Maximization

scholarly journals Submodular Function Maximization

Tractability ◽  
2014 ◽  
pp. 71-104 ◽  
Author(s):  
Andreas Krause ◽  
Daniel Golovin
Keyword(s):  
Submodular Function ◽  
Submodular Function Maximization

scholarly journals An optimal monotone contention resolution scheme for bipartite matchings via a polyhedral viewpoint

2020 ◽  
Author(s):  
Simon Bruggmann ◽  
Rico Zenklusen
Keyword(s):  
Lower Bound ◽  
Submodular Function ◽  
Second Step ◽  
Contention Resolution ◽  
Additional Source ◽  
Submodular Function Maximization ◽  
Versatile Tool ◽  
Resolution Scheme ◽  
Matching Constraints ◽  
Bipartite Case

Abstract Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes round a fractional point by first rounding each coordinate independently, and then dropping some elements to reach a feasible set. Also the second step, where elements are dropped, is typically randomized. This leads to an additional source of randomization within the procedure, which can complicate the analysis. We suggest a different, polyhedral viewpoint to design contention resolution schemes, which avoids to deal explicitly with the randomization in the second step. This is achieved by focusing on the marginals of a dropping procedure. Apart from avoiding one source of randomization, our viewpoint allows for employing polyhedral techniques. Both can significantly simplify the construction and analysis of contention resolution schemes. We show how, through our framework, one can obtain an optimal monotone contention resolution scheme for bipartite matchings, which has a balancedness of 0.4762. So far, only very few results are known about optimality of monotone contention resolution schemes. Our contention resolution scheme for the bipartite case also improves the lower bound on the correlation gap for bipartite matchings. Furthermore, we derive a monotone contention resolution scheme for matchings that significantly improves over the previously best one. More precisely, we obtain a balancedness of 0.4326, improving on a prior 0.1997-balanced scheme. At the same time, our scheme implies that the currently best lower bound on the correlation gap for matchings is not tight. Our results lead to improved approximation factors for various constrained submodular function maximization problems over a combination of matching constraints with further constraints.

scholarly journals Robust monotone submodular function maximization

2018 ◽  
Vol 172 (1-2) ◽  
pp. 505-537 ◽  
Author(s):  
James B. Orlin ◽  
Andreas S. Schulz ◽  
Rajan Udwani
Keyword(s):  
Submodular Function ◽  
Submodular Function Maximization
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