Abstract
Submodular optimization has been well studied in combinatorial optimization. However, there are few works considering about non-submodular optimization problems which also have many applications, such as experimental design, some optimization problems in social networks, etc. In this paper, we consider the maximization of non-submodular function under a knapsack constraint, and explore the performance of the greedy algorithm, which is characterized by the submodularity ratio \(\beta \) and curvature \(\alpha \). In particular, we prove that the greedy algorithm enjoys a tight approximation guarantee of \( (1-e^{-\alpha \beta })/{\alpha }\) for the above problem. To our knowledge, it is the first tight constant factor for this problem. We further utilize illustrative examples to demonstrate the performance of our algorithm.
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Acknowledgements
The first and fifth authors are supported by National Natural Science Foundation of China (No. 11871081). The first author is also supported by the Science and Technology Program of Beijing Education Commission (No. KM201810005006). The second author is supported by National Natural Science Foundation of China (No. 11971447), the Fundamental Research Funds for the Central Universities (No. 201964006), and the Natural Science Foundation of Shandong Province of China (No. ZR2017QA010). The fourth author is supported by National Natural Science Foundation of China (No. 11531014).
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A preliminary version of this paper appeared in Proceedings of the 25th International Computing and Combinatorics Conference, pp. 651–662, 2019.
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Zhang, Z., Liu, B., Wang, Y. et al. Maximizing a monotone non-submodular function under a knapsack constraint. J Comb Optim 43, 1125–1148 (2022). https://doi.org/10.1007/s10878-020-00620-1
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DOI: https://doi.org/10.1007/s10878-020-00620-1