Towards Practical Constrained Monotone Submodular Maximization.

We design new algorithms for maximizing a monotone non-negative submodular function under various constraints, which improve the state-of-the-art in time complexity and/or performance guarantee. We first investigate the cardinality constrained submodular maximization problem that has been widely studied for about four decades. We design an $(1-frac{1}{e}-varepsilon)$-approximation algorithm that makes $O(ncdot max {varepsilon^{-1},loglog k })$ queries. To the best of our knowledge, this is the fastest known algorithm. We further answer the open problem on finding a lower bound on the number of queries. We show that, no (randomized) algorithm can achieve a ratio better than $(frac{1}{2}+Theta(1))$ with $o(frac{n}{log queries. The acceleration above is achieved by our emph{Adaptive Decreasing Threshold} (ADT) algorithm. Based on ADT, we study the $p$-system and $d$ knapsack constrained maximization problem. We show that an $(1/(p+frac{7}{4}d+1)-varepsilon)$-approximate solution can be computed via $O(frac{n}{varepsilon}log frac{n}{varepsilon}max{log frac{1}{varepsilon},loglog n})$ queries. Note that it improves the state of the art in both time complexity and approximation ratio. We also show how to improve the ratio for a single knapsack constraint via $O(ncdot max {varepsilon^{-1},loglog k })$ queries. For maximizing a submodular function with curvature $kappa$ under matroid constraint, we show an $(1-frac{kappa}{e}-varepsilon)$-approximate algorithm that uses $tilde{O}(nk)$ value oracle queries. Our ADT could be utilized to obtain faster algorithms in other problems. To prove our results, we introduce a general characterization between randomized complexity and deterministic complexity of approximation algorithms that could be used in other problems and may be interesting in its own right.