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# Deterministic Approximation for Submodular Maximization over a Matroid in Nearly Linear Time

NIPS 2020, (2020)

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Abstract

We study the problem of maximizing a non-monotone, non-negative submodular function subject to a matroid constraint. The prior best-known deterministic approximation ratio for this problem is $\frac{1}{4}-\epsilon$ under $\mathcal{O}(({n^4}/{\epsilon})\log n)$ time complexity. We show that this deterministic ratio can be improved to $\f...More

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Introduction

- Submodular function maximization has aroused great interests from both academic and industrial societies due to its wide applications such as crowdsourcing [47], information gathering [35], sensor placement [33], influence maximization [37, 48] and exemplar-based clustering [30].
- There appear studies aiming at designing more efficient and practical algorithms for this problem
- In this line of studies, the elegant work of Mirzasoleiman et al [44] and Feldman et al [25] proposes the best deterministic ratio of 1/6 − and the fastest implementation with an expected ratio of 1/4, and their algorithms can handle more general constraints such as a p-set system constraint.
- It is still unclear whether the 1/4 − deterministic ratio for a single matroid constraint can be further improved, or whether there exist faster algorithms achieving the same 1/4 − deterministic ratio

Highlights

- Submodular function maximization has aroused great interests from both academic and industrial societies due to its wide applications such as crowdsourcing [47], information gathering [35], sensor placement [33], influence maximization [37, 48] and exemplar-based clustering [30]
- Our approach is based on a novel algorithmic framework of simultaneously constructing two candidate solution sets through greedy search, which enables us to get improved performance bounds by fully exploiting the properties of independence systems
- We generate Barabási–Albert (BA) graphs for comparison, and the results are qualitatively similar. It can be seen from Fig. 2(a) that Fantom incurs the largest number of queries to objective function, as it leverages repeated greedy searching processes to find a solution with good quality
- We have proposed the first deterministic algorithm to achieve an approximation ratio of 1/4 for maximizing a non-monotone, non-negative submodular function subject to a matroid constraint, and our algorithm can be accelerated to achieve nearly-linear running time
- In contrast to the existing algorithms adopting the “repeated greedy-search” framework proposed by [31], our algorithms are designed based on a novel “simultaneous greedy-search” framework, where two candidate solutions are constructed simultaneously, and a pair of an element and a candidate solution is greedily selected at each step to maximize the marginal gain
- We have evaluated the performance of our algorithms in two concrete applications for social network monitoring and multi-product viral marketing, and the extensive experimental results demonstrate that our algorithms runs in orders of magnitude faster than the state-of-the-art algorithms, while achieving approximately the same utility

Results

- The experimental results are shown in Fig. 2.
- The authors generate Barabási–Albert (BA) graphs for comparison, and the results are qualitatively similar
- It can be seen from Fig. 2(a) that Fantom incurs the largest number of queries to objective function, as it leverages repeated greedy searching processes to find a solution with good quality.
- TwinGreedyFast significantly outperforms Fantom, RRG and SampleGreedy by more than an order of magnitude in Fig. 2(a) and Fig. 2(b), as it achieves nearly linear time complexity
- It can be seen from Figs.
- Fig. 2(c) shows that all the implemented algorithms achieve approximately the same utility on the ER random graph

Conclusion

- The authors have proposed the first deterministic algorithm to achieve an approximation ratio of 1/4 for maximizing a non-monotone, non-negative submodular function subject to a matroid constraint, and the algorithm can be accelerated to achieve nearly-linear running time.
- The authors' algoirthms can be directly used to handle a more general p-set system constraint or monotone submodular functions, while still achieving nice performance bounds.
- The authors have evaluated the performance of the algorithms in two concrete applications for social network monitoring and multi-product viral marketing, and the extensive experimental results demonstrate that the algorithms runs in orders of magnitude faster than the state-of-the-art algorithms, while achieving approximately the same utility

Summary

## Introduction:

Submodular function maximization has aroused great interests from both academic and industrial societies due to its wide applications such as crowdsourcing [47], information gathering [35], sensor placement [33], influence maximization [37, 48] and exemplar-based clustering [30].- There appear studies aiming at designing more efficient and practical algorithms for this problem
- In this line of studies, the elegant work of Mirzasoleiman et al [44] and Feldman et al [25] proposes the best deterministic ratio of 1/6 − and the fastest implementation with an expected ratio of 1/4, and their algorithms can handle more general constraints such as a p-set system constraint.
- It is still unclear whether the 1/4 − deterministic ratio for a single matroid constraint can be further improved, or whether there exist faster algorithms achieving the same 1/4 − deterministic ratio
## Results:

The experimental results are shown in Fig. 2.- The authors generate Barabási–Albert (BA) graphs for comparison, and the results are qualitatively similar
- It can be seen from Fig. 2(a) that Fantom incurs the largest number of queries to objective function, as it leverages repeated greedy searching processes to find a solution with good quality.
- TwinGreedyFast significantly outperforms Fantom, RRG and SampleGreedy by more than an order of magnitude in Fig. 2(a) and Fig. 2(b), as it achieves nearly linear time complexity
- It can be seen from Figs.
- Fig. 2(c) shows that all the implemented algorithms achieve approximately the same utility on the ER random graph
## Conclusion:

The authors have proposed the first deterministic algorithm to achieve an approximation ratio of 1/4 for maximizing a non-monotone, non-negative submodular function subject to a matroid constraint, and the algorithm can be accelerated to achieve nearly-linear running time.- The authors' algoirthms can be directly used to handle a more general p-set system constraint or monotone submodular functions, while still achieving nice performance bounds.
- The authors have evaluated the performance of the algorithms in two concrete applications for social network monitoring and multi-product viral marketing, and the extensive experimental results demonstrate that the algorithms runs in orders of magnitude faster than the state-of-the-art algorithms, while achieving approximately the same utility

- Table1: Approximation for Non-monotone Submodular Maximization over a Matroid
- Table2: The utility of TwinGreedyFast (×105) vs. the parameter (BA, h = 5)

Related work

- When the considered submodular function f (·) is monotone, Calinescu et al [15] propose an optimal 1 − 1/e expected ratio for the problem of submodular maximization subject to a matroid constraint (SMM). The SMM problem seems to be harder when f (·) is non-monotone, and the current bestknown expected ratio is 0.385 [10], got after a series of studies [24, 29, 38, 50]. However, all these approaches are based on tools with high time complexity such as multilinear extension.

There also exist efficient deterministic algorithms for the SMM problem: Gupta et al [31] are the first to apply the “repeated greedy search” framework described in last section and achieve 1/12 − ratio, which is improved to 1/6 − by Mirzasoleiman et al [44] and Feldman et al [25]; under a more general p-set system constraint, Mirzasoleiman et al. [44]

achieve p (p+1)(2p+1)

deterministic ratio and Feldman et al [25] achieve p+2√1 p+3 − deterministic ratio (assuming that they use the USM algorithm with 1/2 − deterministic ratio in [9]); some studies also propose streaming algorithms under various constraints [32, 45].

As regards the efficient randomized algorithms for the SMM problem, the SampleGreedy algorithm in [25] achieves 1/4 expected ratio with O(nr) running time; the algorithms in [11] also achieve a 1/4 expected ratio with slightly worse time complexity of O(nr log n) and a 0.283 expected ratio under cubic time complexity of O(nr log n + r3+ )1; Chekuri and Quanrud [16] provide a 0.172 − √expected ratio under O(log n log r/ 2) adaptive rounds; and Feldman et al [26] propose a 1/(3 + 2 2) expected ratio under the steaming setting. It is also noted that Buchbinder et al [14] provide a de-randomized version of the algorithm in [11] for monotone submodular maximization, which has time complexity of O(nr2). However, it remains an open problem to find the approximation ratio of this de-randomized algorithm for the SMM problem with a non-monotone objective function.

Funding

- This work was supported by the National Key R&D Program of China under Grant No 2018AAA0101204, the National Natural Science Foundation of China (NSFC) under Grant No 61772491 and Grant No U1709217, Anhui Initiative in Quantum Information Technologies under Grant No AHY150300, and the Fundamental Research Funds for the Central Universities

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