Fast submodular maximization subject to k-extendible system constraints
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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-ϵ) for monotone submodular functions and of (p(1-p)-ϵ) for non-monotone cases with expected computational complexity of only O(pn/ϵr/ϵ), where r is the largest size of the feasible solutions, 0<p ≤1/1+k is the sampling probability and 0< ϵ < p. If we fix the sampling probability p as 1/1+k, we get the best approximation ratios for both monotone and non-monotone submodular functions which are (1/1+k-ϵ) and (k/(1+k)^2-ϵ) respectively. While the parameter ϵ 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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