Abstract
Two-stage submodular maximization problems have been recently applied in machine learning, economics and engineering. In this paper, we consider a two-stage submodular problem subject to cardinality constraint and matroid constraint. Previous work for this problem usually assume that the objective functions are non-negative and monotone. Our focus in this work relaxes these assumptions by considering an objective function which is the expected difference of a non-negative monotone submodular function and a non-negative monotone modular function, and hence neither non-negative nor monotone. We present strong approximation guarantees by offering two bi-factor approximation algorithms for this problem. The first is a deterministic \(\left( \frac{1}{2}\left( 1-e^{-2}\right) , 1\right) \)-approximation algorithm, and the second is a randomized \(\left( \frac{1}{2}\left( 1-e^{-2}\right) -\epsilon , 1\right) \)-approximation algorithm with improved time efficiency. Moreover, we generalize the matroid constraint to k-matroid constraint and also give the corresponding approximation algorithms.
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Acknowledgements
This research is supported or partially supported by the National Natural Science Foundation of China (Grant Nos. 11871280, 11501171, 11771251, 11971349, 11771386 and 11728104), the Natural Sciences and Engineering Research Council of Canada (NSERC) Grant 06446 and Qinglan Project.
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Liu, Z., Chang, H., Ma, R., Du, D., Zhang, X. (2020). Two-Stage Submodular Maximization Problem Beyond Non-negative and Monotone. In: Chen, J., Feng, Q., Xu, J. (eds) Theory and Applications of Models of Computation. TAMC 2020. Lecture Notes in Computer Science(), vol 12337. Springer, Cham. https://doi.org/10.1007/978-3-030-59267-7_13
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