Streaming Algorithms for Maximizing Monotone Submodular Functions Under a Knapsack Constraint

In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access only to a small fraction of the data stored in primary memory. For this problem, we propose a (0.363-epsilon)-approximation algorithm, requiring only a single pass through the data; moreover, we propose a (0.4-epsilon)-approximation algorithm requiring a constant number of passes through the data. The required memory space of both algorithms depends only on the size of the knapsack capacity and epsilon.

[1]  T.-H. Hubert Chan,et al.  Online Submodular Maximization with Free Disposal: Randomization Beats ¼ for Partition Matroids , 2017, SODA.

[2]  T.-H. Hubert Chan,et al.  Online Submodular Maximization Problem with Vector Packing Constraint , 2017, ESA.

[3]  Shuguang Cui,et al.  Submodular maximization with multi-knapsack constraints and its applications in scientific literature recommendations , 2016, 2016 IEEE Global Conference on Signal and Information Processing (GlobalSIP).

[4]  L. Wolsey.  Maximising Real-Valued Submodular Functions: Primal and Dual Heuristics for Location Problems , 1982, Math. Oper. Res..

[5]  Jan Vondrák,et al.  Maximizing a Monotone Submodular Function Subject to a Matroid Constraint , 2011, SIAM J. Comput..

[6]  Yuichi Yoshida,et al.  Maximizing a Monotone Submodular Function with a Bounded Curvature under a Knapsack Constraint , 2016, SIAM J. Discret. Math..

[7]  Hui Lin,et al.  Multi-document Summarization via Budgeted Maximization of Submodular Functions , 2010, NAACL.

[8]  Maxim Sviridenko,et al.  A note on maximizing a submodular set function subject to a knapsack constraint , 2004, Oper. Res. Lett..

[9]  Noga Alon,et al.  Optimizing budget allocation among channels and influencers , 2012, WWW.

[10]  Vahab S. Mirrokni,et al.  Almost Optimal Streaming Algorithms for Coverage Problems , 2016, SPAA.

[11]  Yuval Filmus,et al.  Monotone Submodular Maximization over a Matroid via Non-Oblivious Local Search , 2012, SIAM J. Comput..

[12]  Andreas Krause,et al.  Streaming Non-monotone Submodular Maximization: Personalized Video Summarization on the Fly , 2017, AAAI.

[13]  Huy L. Nguyen,et al.  A Nearly-linear Time Algorithm for Submodular Maximization with a Knapsack Constraint , 2019, ICALP.

[14]  Andreas Krause,et al.  Streaming submodular maximization: massive data summarization on the fly , 2014, KDD.

[15]  Amit Chakrabarti,et al.  Submodular maximization meets streaming: matchings, matroids, and more , 2013, Math. Program..

[16]  Huy L. Nguyen,et al.  The Power of Randomization: Distributed Submodular Maximization on Massive Datasets , 2015, ICML.

[17]  Ken-ichi Kawarabayashi,et al.  Optimal Budget Allocation: Theoretical Guarantee and Efficient Algorithm , 2014, ICML.

[18]  Kent Quanrud,et al.  Submodular Function Maximization in Parallel via the Multilinear Relaxation , 2019, SODA.

[19]  Huy L. Nguyen,et al.  Submodular Maximization with Nearly-optimal Approximation and Adaptivity in Nearly-linear Time , 2018, SODA.

[20]  Amin Karbasi,et al.  Do Less, Get More: Streaming Submodular Maximization with Subsampling , 2018, NeurIPS.

[21]  Kent Quanrud,et al.  Streaming Algorithms for Submodular Function Maximization , 2015, ICALP.

[22]  Eric Balkanski,et al.  The adaptive complexity of maximizing a submodular function , 2018, STOC.

[23]  J. Vondrák,et al.  Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes , 2014 .

[24]  Naonori Kakimura,et al.  Multi-Pass Streaming Algorithms for Monotone Submodular Function Maximization , 2018, Theory Comput. Syst..

[25]  Huy L. Nguyen,et al.  A New Framework for Distributed Submodular Maximization , 2015, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[26]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[27]  Andreas Krause,et al.  Near-Optimal Sensor Placements in Gaussian Processes: Theory, Efficient Algorithms and Empirical Studies , 2008, J. Mach. Learn. Res..

[28]  Hadas Shachnai,et al.  Maximizing submodular set functions subject to multiple linear constraints , 2009, SODA.

[29]  Hui Lin,et al.  A Class of Submodular Functions for Document Summarization , 2011, ACL.

[30]  Shuguang Cui,et al.  Streaming Algorithms for News and Scientific Literature Recommendation: Submodular Maximization with a d-Knapsack Constraint , 2016, ArXiv.

[31]  Jon Lee,et al.  Maximum Entropy SamplingBased in part on the article “Maximum entropy sampling” by Jon Lee, which appeared in the Encyclopedia of Environmetrics. , 2013 .

[32]  Jan Vondrák,et al.  Submodular Maximization over Multiple Matroids via Generalized Exchange Properties , 2009, Math. Oper. Res..

[33]  Jan Vondrák,et al.  Fast algorithms for maximizing submodular functions , 2014, SODA.

[34]  Yuval Filmus,et al.  A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[35]  Sergei Vassilvitskii,et al.  Fast greedy algorithms in mapreduce and streaming , 2013, SPAA.

[36]  Andrew McGregor,et al.  Graph stream algorithms: a survey , 2014, SGMD.

[37]  Andrew McGregor,et al.  Better Streaming Algorithms for the Maximum Coverage Problem , 2018, Theory of Computing Systems.

[38]  Jon Lee,et al.  Maximum-Entropy Sampling , 2006, Springer Series in Operations Research and Financial Engineering.

[39]  Eric Balkanski,et al.  An Exponential Speedup in Parallel Running Time for Submodular Maximization without Loss in Approximation , 2018, SODA.

[40]  Éva Tardos,et al.  Maximizing the Spread of Influence through a Social Network , 2015, Theory Comput..