Near-optimal Approximate Discrete and Continuous Submodular Function Minimization
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In this paper we provide improved running times and oracle complexities for approximately minimizing a submodular function. Our main result is a randomized algorithm, which given any submodular function defined on n-elements with range [-1, 1], computes an ϵ-additive approximate minimizer in Õ(n/ϵ^2) oracle evaluations with high probability. This improves over the Õ(n^5/3/ϵ^2) oracle evaluation algorithm of Chakrabarty (STOC 2017) and the Õ(n^3/2/ϵ^2) oracle evaluation algorithm of Hamoudi . Further, we leverage a generalization of this result to obtain efficient algorithms for minimizing a broad class of nonconvex functions. For any function f with domain [0, 1]^n that satisfies ∂^2f/∂ x_i ∂ x_j< 0 for all i ≠ j and is L-Lipschitz with respect to the L^∞-norm we give an algorithm that computes an ϵ-additive approximate minimizer with Õ(n ·poly(L/ϵ)) function evaluation with high probability.
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