Approximation guarantees for parallelized maximization of monotone non-submodular function with a cardinality constraint

Min Cui (), Dachuan Xu (), Longkun Guo () and Dan Wu ()
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Min Cui: Beijing University of Technology
Dachuan Xu: Beijing University of Technology
Longkun Guo: Qilu University of Technology (Shandong Academy of Sciences)
Dan Wu: Henan University of Science and Technology

Journal of Combinatorial Optimization, 2022, vol. 43, issue 5, No 36, 1690 pages

Abstract: 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 $$\gamma $$ γ to connect the monotone set function with submodularity. The algorithm achieves an approximation ratio of $$1-e^{-\gamma ^2}-\varepsilon $$ 1 - e - γ 2 - ε and consumes $$O(\log (n/\eta )/\varepsilon ^2)$$ O ( log ( n / η ) / ε 2 ) adaptive rounds and $$O(n\log \log (k)/\varepsilon ^3)$$ O ( n log log ( k ) / ε 3 ) oracle queries in expectation. Furthermore, when $$\gamma =1$$ γ = 1 , the algorithm achieves an approximation guarantee $$1-1/e-\varepsilon $$ 1 - 1 / e - ε , achieving the same ratio as the state-of-art result for the submodular version of the problem.

Keywords: Non-submodular optimization; Cardinality constraint; Submodularity ratio; Parallel algorithm (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10878-021-00719-z

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