Approximation Algorithms for the k-Median Problem

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.

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Approximation Algorithms for Maximin Fair Division

ACM Transactions on Economics and Computation ◽

10.1145/3381525 ◽

2020 ◽

Vol 8 (1) ◽

pp. 1-28

Author(s):

Siddharth Barman ◽

Sanath Kumar Krishnamurthy

Keyword(s):

Approximation Algorithms ◽

Fair Division

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Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint

Proceedings of the ACM on Measurement and Analysis of Computing Systems ◽

10.1145/3447386 ◽

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.

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