Abstract
Emerging applications in machine learning have imposed the problem of monotone non-submodular maximization subject to a cardinality constraint. Meanwhile, parallelism is prevalent for large-scale optimization problems in bigdata scenario while adaptive complexity is an important measurement of parallelism since it quantifies the number of sequential rounds by which the multiple independent functions can be evaluated in parallel. For a monotone non-submodular function and a cardinality constraint, this paper devises an adaptive algorithm for maximizing the function value with the cardinality constraint through employing the generic submodularity ratio \(\gamma \) to connect the monotone set function with submodularity. The algorithm achieves an approximation ratio of \(1-e^{-\gamma ^2}-\varepsilon \) and consumes \(O(\log (n/\eta )/\varepsilon ^2)\) adaptive rounds and \(O(n\log \log (k)/\varepsilon ^3)\) oracle queries in expectation. Furthermore, when \(\gamma =1\), the algorithm achieves an approximation guarantee \(1-1/e-\varepsilon \), achieving the same ratio as the state-of-art result for the submodular version of the problem.
Similar content being viewed by others
References
Agarwal A, Assadi S, Khanna S (2019) Stochastic submodular cover with limited adaptivity. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 323–342
Alon N, Spencer J (2000) The probabilistic method. Wiley, New York
Attigeri G, Manohara PMM, Radhika MP (2019) Feature selection using submodular approach for financial big data. J Inf Process Syst 15(6):1306–1325
Balkanski E, Rubinstein A, Singer Y (2019) An exponential speedup in parallel running time for submodular maximization without loss in approximation. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 283–302
Balkanski E, Singer Y (2018) The adaptive complexity of maximizing a submodular function. In: Proceedings of the 50th annual ACM SIGACT symposium on theory of computing, pp 1138–1151
Balkanski E, Singer Y (2020) A lower bound for parallel submodular minimization. In: Proceedings of the 52nd annual ACM SIGACT symposium on theory of computing, pp 130–139
Bian AA, Buhmann JM, Krause A, Tschiatschek S (2017) Guarantees for greedy maximization of non-submodular functions with applications. In: Proceedings of the 34th international proceedings on international conference on machine learning, vol 70, pp 498–507
Breuer A, Balkanski E, Singer Y (2020) The FAST algorithm for submodular maximization. In: Proceedings of the 37th international conference on machine learning, pp 1134–1143
Buchbinder N, Feldman M, Naor J, Schwartz R (2015) A tight linear time (1/2)-approximation for unconstrained submodular maximization. SIAM J Comput 44(5):1384–1402
Chekuri C, Quanrud K (2019a) Submodular function maximization in parallel via the multilinear relaxation. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 303–322
Chekuri C, Quanrud K (2019b) Parallelizing greedy for submodular set function maximization in matroids and beyond. In: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing, pp 78–89
Chernoff H (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann Math Stat 23(4):493–507
Ene A, Nguyen HL (2019) Submodular maximization with nearly-optimal approximation and adaptivity in nearly-linear time. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 274–282
Ene A, Nguyn HL, Vladu A (2019) Submodular maximization with matroid and packing constraints in parallel. In: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing, pp 90–101
Ene A, Nguyn HL (2020) Parallel algorithm for non-monotone DR-submodular maximization. In: Proceedings of the 37th international conference on machine learning, pp 2902–2911
Fahrbach M, Miller GL, Peng R, Sawlani S, Wang J, Xu SC (2018) Graph sketching against adaptive adversaries applied to the minimum degree algorithm. In: Proceedings of the 59th annual symposium on foundations of computer science, pp 101–112
Fahrbach M, Mirrokni V, Zadimoghaddam M (2019) Submodular maximization with nearly optimal approximation, adaptivity and query complexity. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 255–273
Feige U, Mirrokni V, Vondrak J (2011) Maximizing non-monotone submodular functions. SIAM J Comput 40(4):1133–1153
Golovin D, Krause A (2010) Adaptive submodularity: theory and applications in active learning and stochastic optimization. J Artif Intell Res 42(1):427–486
Gong S, Nong Q, Liu W, Fang Q (2019) Parametric monotone function maximization with matroid constraints. J Global Optim 75(3):833–849
Kawahara Y, Nagano K, Okamoto Y (2011) Submodular fractional programming for balanced clustering. Pattern Recogn Lett 42(2):235–243
Kempe D, Kleinberg J, Tardos E (2015) Maximizing the spread of influence through a social network. Theory Comput 11(1):105–147
Kuhnle A, Smith J, Crawford VG, Thai MT (2018) Fast maximization of non-submodular, monotonic functions on the integer lattice. In: Proceedings of the 35th international proceedings on international conference on machine learning, pp 2786–2795
Kuhnle A (2021) Nearly linear-time, parallelizable algorithms for non-monotone submodular maximization. In: Proceedings of the 35th AAAI conference on artificial intelligence
Lin H, Bilmes JA (2011) A class of submodular functions for document summarization. In: Proceedings of the 49th annual meeting of the association for computational linguistics: human language technologies, vol 1, pp 510–520
Lin Y, Chen W, Lui JC (2017) Boosting information spread: an algorithmic approach. In: Proceedings of the international conference on data engineering, pp 883–894
Mirrokni V, Zadimoghaddam M (2015) Randomized composable core-sets for distributed submodular maximization. In: Proceedings of the 47th annual ACM symposium on theory of computing, pp 153–162
Nemhauser GL, Wolsey LA, Fisher ML (1978) An analysis of approximations for maximizing submodular set functions I. Math Program 14(1):265–294
Nemhauser GL, Wolsey LA (1978) Best algorithms for approximating the maximum of a submodular set function. Math Oper Res 3(3):177–188
Nong Q, Sun T, Gong S, Fang Q, Du D, Shao X (2019) Maximize a monotone function with a generic submodularity ratio. In: Proceedings of the international proceedings on algorithmic applications in management, pp 249–260
Pan X, Jegelka S, Gonzalez JE, Bradley JK, Jordan MI (2014) Parallel double greedy submodular maximization. In: Proceedings of the 27th international conference on neural information processing systems, vol 1, pp 118–126
Parambath SA, Chawla S, Vijayakumar N (2018) SAGA: a submodular greedy algorithm for group recommendation. In: Proceedings of international conference on international conference on artificial intelligence, pp 3900–3908
Tang S (2021) Beyond pointwise submodularity: non-monotone adaptive submodular maximization in linear time. Theor Comput Sci 850:249–261
Wei K, Iyer R, Bilmes JA (2015) Submodularity in data subset selection and active learning. In: Proceedings of the 32nd international conference on international conference on machine learning, vol 37, pp 1954–1963
Zhang Z, Liu B, Wang Y, Xu D, Zhang D (2019) Greedy algorithm for maximization of non-submodular functions subject to knapsack constraint. In: Proceedings of the international proceedings on computing and combinatorics conference, pp 651–662
Acknowledgements
The first two authors are supported by National Natural Science Foundation of China (No. 11871081) and Bei**g Natural Science Foundation Project (No. Z200002). The third author is supported by National Natural Science Foundation of China (No. 61772005) and Outstanding Youth Innovation Team Project for Universities of Shandong Province (No. 2020KJN008). The fourth author is supported by the National Natural Science Foundation of China (No. 11701150).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
A preliminary version of this paper appeared in Proceedings of the 14th International Conference on Algorithmic Aspects in Information and Management, pp. 195–203, 2020.
Rights and permissions
About this article
Cite this article
Cui, M., Xu, D., Guo, L. et al. Approximation guarantees for parallelized maximization of monotone non-submodular function with a cardinality constraint. J Comb Optim 43, 1671–1690 (2022). https://doi.org/10.1007/s10878-021-00719-z
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-021-00719-z