Fast submodular maximization subject to k-extendible system constraints

As the scales of data sets expand rapidly in some application scenarios, increasing efforts have been made to develop fast submodular maximization algorithms. This paper presents a currently the most efficient algorithm for maximizing general non-negative submodular objective functions subject to $k$-extendible system constraints. Combining the sampling process and the decreasing threshold strategy, our algorithm Sample Decreasing Threshold Greedy Algorithm (SDTGA) obtains an expected approximation guarantee of ($p-\epsilon$) for monotone submodular functions and of ($p(1-p)-\epsilon$) for non-monotone cases with expected computational complexity of only $O(\frac{pn}{\epsilon}\ln\frac{r}{\epsilon})$, where $r$ is the largest size of the feasible solutions, $0<p \leq \frac{1}{1+k}$ is the sampling probability and $0< \epsilon < p$. If we fix the sampling probability $p$ as $\frac{1}{1+k}$, we get the best approximation ratios for both monotone and non-monotone submodular functions which are $(\frac{1}{1+k}-\epsilon)$ and $(\frac{k}{(1+k)^{2}-\epsilon)$} respectively. While the parameter $\epsilon$ exists for the trade-off between the approximation ratio and the time complexity. Therefore, our algorithm can handle larger scale of submodular maximization problems than existing algorithms.