Practical Budgeted Submodular Maximization

We consider the problem of maximizing a non-negative monotone submodular function subject to knapsack constraint, which is also known as Budgeted Submodular Maximization (BSM) problem. Sviridenko (Operat Res Lett 32:41–43, 2004) showed that by guessing 3 appropriate elements an optimal solution, and then executing greedy algorithm, one can obtain approximation ratio $$\alpha =1-{1}/{e} \approx 0.632$$ for BSM. However, need guess (by enumeration) makes algorithm impractical it leads time complexity roughly $$O(n^5)$$ (this be slightly improved using thresholding technique Badanidiyuru & Vondrák (in: SODA, 1497–1514, 2014) but only $$O(n^4)$$ ). Our main results in this paper show fewer guesses suffice. Specifically, making 2 (and 2014), we get same $$ with $$O(n^3)$$ . Furthermore, single guess, almost good $$0.6174 > 0.9767\alpha $$O(n^2)$$ time. Prior our work, algorithms were close BSM Ene Nguyen ICALP, 53:1–53:12, 2019) achieves $$(\alpha -\varepsilon )$$ -approximation. requires $${(1/\varepsilon )}^{O(1/\varepsilon ^4)}n\log ^2 n$$ time, hence, theoretical interest since ^4)}$$ huge even moderate values $$\varepsilon In contrast, all analyze are simple parallelizable, them candidates practical use. Recently, Tang et al. Proc ACM Meas Anal Comput Syst, 5(1): 08:1–08:22, 2021) studied already has long research history, proved admits at least 0.405 (without any guesses). The last part improves over result shows within range [0.427, 0.462].