Fast submodular maximization subject to k-extendible system constraints


Abstract in English

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 $O(frac{pn}{epsilon}lnfrac{r}{epsilon})$, 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.

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