Deletion-Robust Submodular Maximization under a Knapsack Constraint

In recent years, as one of the important problems in machine learning optimization theory, submodular maximization has received extensive research attention. Due to the property of diminishing returns of submodular function, many natural problems can be cast to the submodular maximization problem and that can be addressed relatively efficiently. In this paper, we investigate the deletion robust sub modular maximization problem under the knapsack constraint. The robustness means the algorithm can guarantee the quality of its output even after an adversary deleted some elements from the ground set $V$ . During the period when $V$ is accessed, the algorithm does not know the elements the adversary deleted. We consider the scenario where the adversary deletes elements randomly, which is consistent with unexpected events in nature. For offline and streaming scenarios, we present two algorithms that achieve an approximation ratio of $(\frac{1}{4} - \epsilon)$ in expectation and a summary size of $O (k+ \frac{d\log^{2}B}{\epsilon^{3}})$ and $O(\frac{k\log B}{\epsilon}+\frac{d\log^{3} B}{\epsilon^{4}})$ respectively, where $k$ represents the maximum element number of any feasible solution with budget $B$ and $d$ is the maximum number of deleted elements. We conduct experiments on real-world applications to evaluate our proposed algorithms and the experimental results confirm the superiority of our proposed algorithms.