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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

# 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
Tables
• 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
Reference
• Z. Abbassi, V. S. Mirrokni, and M. Thakur. Diversity maximization under matroid constraints. In KDD, pages 32–40, 2013.
• A. Badanidiyuru and J. Vondrák. Fast algorithms for maximizing submodular functions. In SODA, pages 1497–1514, 2014.
• M.-F. Balcan and N. J. Harvey. Submodular functions: Learnability, structure, and optimization. SIAM Journal on Computing, 47(3):703–754, 2018.
• E. Balkanski, A. Breuer, and Y. Singer. Non-monotone submodular maximization in exponentially fewer iterations. In NIPS, pages 2353–2364, 2018.
• E. Balkanski, A. Rubinstein, and Y. Singer. An optimal approximation for submodular maximization under a matroid constraint in the adaptive complexity model. In STOC, pages 66–77, 2019.
• N. Barbieri, F. Bonchi, and G. Manco. Topic-aware social influence propagation models. In ICDM, pages 81–90, 2012.
• C. Borgs, M. Brautbar, J. Chayes, and B. Lucier. Maximizing social influence in nearly optimal time. In SODA, pages 946–957, 2014.
• V. Bryant and H. Perfect. Independence Theory in Combinatorics: An Introductory Account with Applications to Graphs and Transversals. Springer, 1980.
• N. Buchbinder and M. Feldman. Deterministic algorithms for submodular maximization problems. ACM Transactions on Algorithms, 14(3):1–20, 2018.
• N. Buchbinder and M. Feldman. Constrained submodular maximization via a nonsymmetric technique. Mathematics of Operations Research, 44(3):988–1005, 2019.
• N. Buchbinder, M. Feldman, J. Naor, and R. Schwartz. Submodular maximization with cardinality constraints. In SODA, pages 1433–1452, 2014.
• N. Buchbinder, M. Feldman, J. Seffi, and R. Schwartz. A tight linear time (1/2)-approximation for unconstrained submodular maximization. SIAM Journal on Computing, 44(5):1384–1402, 2015.
• N. Buchbinder, M. Feldman, and R. Schwartz. Comparing apples and oranges: Query trade-off in submodular maximization. Mathematics of Operations Research, 42(2):308–329, 2017.
• N. Buchbinder, M. Feldman, and M. Garg. Deterministic (1/2+ ε)-approximation for submodular maximization over a matroid. In SODA, pages 241–254, 2019.
• G. Calinescu, C. Chekuri, M. Pal, and J. Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40(6):1740–1766, 2011.
• C. Chekuri and K. Quanrud. Parallelizing greedy for submodular set function maximization in matroids and beyond. In STOC, pages 78–89, 2019.
• W. Chen, Y. Wang, and S. Yang. Efficient influence maximization in social networks. In KDD, pages 199–208, 2009.
• W. Chen, W. Zhang, and H. Zhao. Gradient method for continuous influence maximization with budget-saving considerations. In AAAI, 2020.
• M. Corah and N. Michael. Distributed submodular maximization on partition matroids for planning on large sensor networks. In CDC, pages 6792–6799, 2018.
• K. El-Arini and C. Guestrin. Beyond keyword search: Discovering relevant scientific literature. In KDD, pages 439–447, 2011.
• K. El-Arini, G. Veda, D. Shahaf, and C. Guestrin. Turning down the noise in the blogosphere. In KDD, pages 289–298, 2009.
• A. Ene and H. L. Nguyen. Towards nearly-linear time algorithms for submodular maximization with a matroid constraint. In ICALP, page 54:1–54:14, 2019.
• M. Fahrbach, V. Mirrokni, and M. Zadimoghaddam. Non-monotone submodular maximization with nearly optimal adaptivity and query complexity. In ICML, pages 1833–1842, 2019.
• M. Feldman, J. Naor, and R. Schwartz. A unified continuous greedy algorithm for submodular maximization. In FOCS, pages 570–579, 2011.
• M. Feldman, C. Harshaw, and A. Karbasi. Greed is good: Near-optimal submodular maximization via greedy optimization. In COLT, pages 758–784, 2017.
• M. Feldman, A. Karbasi, and E. Kazemi. Do less, get more: Streaming submodular maximization with subsampling. In NIPS, pages 732–742, 2018.
• M. Fisher, G. Nemhauser, and L. Wolsey. An analysis of approximations for maximizing submodular set functions—ii. Mathematical Programming Study, 8:73–87, 1978.
• K. Fujii and S. Sakaue. Beyond adaptive submodularity: Approximation guarantees of greedy policy with adaptive submodularity ratio. In ICML, pages 2042–2051, 2019.
• S. O. Gharan and J. Vondrák. Submodular maximization by simulated annealing. In SODA, pages 1098–1116, 2011.
• R. Gomes and A. Krause. Budgeted nonparametric learning from data streams. In ICML, page 391–398, 2010.
• A. Gupta, A. Roth, G. Schoenebeck, and K. Talwar. Constrained non-monotone submodular maximization: Offline and secretary algorithms. In WINE, pages 246–257, 2010.
• R. Haba, E. Kazemi, M. Feldman, and A. Karbasi. Streaming submodular maximization under a k-set system constraint. In ICML, (arXiv:2002.03352), 2020.
• R. K. Iyer and J. A. Bilmes. Submodular optimization with submodular cover and submodular knapsack constraints. In NIPS, pages 2436–2444, 2013.
• D. Kempe, J. Kleinberg, and É. Tardos. Maximizing the spread of influence through a social network. In KDD, pages 137–146, 2003.
• A. Krause and C. Guestrin. Submodularity and its applications in optimized information gathering. ACM Transactions on Intelligent Systems and Technology, 2(4):1–20, 2011.
• A. Kuhnle. Interlaced greedy algorithm for maximization of submodular functions in nearly linear time. In NIPS, pages 2371–2381, 2019.
• A. Kuhnle, J. D. Smith, V. Crawford, and M. Thai. Fast maximization of non-submodular, monotonic functions on the integer lattice. In ICML, pages 2786–2795, 2018.
• J. Lee, V. S. Mirrokni, V. Nagarajan, and M. Sviridenko. Maximizing nonmonotone submodular functions under matroid or knapsack constraints. SIAM Journal on Discrete Mathematics, 23 (4):2053–2078, 2010.
• J. Leskovec and A. Krevl. Snap datasets: Stanford large network dataset collection. URL: http://snap.stanford.edu/, 2014.
• J. Leskovec, A. Krause, C. Guestrin, C. Faloutsos, J. VanBriesen, and N. Glance. Cost-effective outbreak detection in networks. In KDD, pages 420–429, 2007.
• H. Lin and J. Bilmes. A class of submodular functions for document summarization. In ACL/HLT, pages 510–520, 2011.
• M.-Y. Liu, O. Tuzel, S. Ramalingam, and R. Chellappa. Entropy-rate clustering: Cluster analysis via maximizing a submodular function subject to a matroid constraint. IEEE Transactions on Pattern Analysis and Machine Intelligence, 36(1):99–112, 2014.
• B. Mirzasoleiman, A. Badanidiyuru, A. Karbasi, J. Vondrák, and A. Krause. Lazier than lazy greedy. In AAAI, pages 1812–1818, 2015.
• B. Mirzasoleiman, A. Badanidiyuru, and A. Karbasi. Fast constrained submodular maximization: Personalized data summarization. In ICML, pages 1358–1367, 2016.
• B. Mirzasoleiman, S. Jegelka, and A. Krause. Streaming non-monotone submodular maximization: Personalized video summarization on the fly. In AAAI, pages 1379–1386, 2018.
• A. Singla, I. Bogunovic, G. Bartok, A. Karbasi, and A. Krause. Near-optimally teaching the crowd to classify. In ICML, pages 154–162, 2014.
• A. Singla, S. Tschiatschek, and A. Krause. Noisy submodular maximization via adaptive sampling with applications to crowdsourced image collection summarization. In AAAI, page 2037–2041, 2016.
• T. Soma and Y. Yoshida. Non-monotone dr-submodular function maximization. In AAAI, pages 898–904, 2017.
• T. Soma and Y. Yoshida. Maximizing monotone submodular functions over the integer lattice. Mathematical Programming, 172(1-2):539–563, 2018.
• J. Vondrák. Symmetry and approximability of submodular maximization problems. SIAM Journal on Computing, 42(1):265–304, 2013.
• R. K. Williams, A. Gasparri, and G. Ulivi. Decentralized matroid optimization for topology constraints in multi-robot allocation problems. In ICRA, pages 293–300, 2017.
• J. Xu, L. Mukherjee, Y. Li, J. Warner, J. M. Rehg, and V. Singh. Gaze-enabled egocentric video summarization via constrained submodular maximization. In CVPR, pages 2235–2244, 2015.
Author
zongmai Cao
Shuang Cui
Benwei Wu