Fast Evolutionary Algorithms for Maximization of Cardinality-Constrained Weakly Submodular Functions
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We study the monotone, weakly submodular maximization problem (WSM), which is to find a subset of size k from a universe of size n that maximizes a monotone, weakly submodular objective function f. For objectives with submodularity ratio γ, we provide two novel evolutionary algorithms that have an expected approximation guarantee of (1-n^-1)(1-e^-γ-ϵ) for WSM in linear time, improving upon the cubic time complexity of previous evolutionary algorithms for this problem. This improvement is a result of restricting mutations to local changes, a biased random selection of which set to mutate, and an improved theoretical analysis. In the context of several applications of WSM, we demonstrate the ability of our algorithms to quickly exceed the solution of the greedy algorithm and converge faster than existing evolutionary algorithms for WSM.
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