Near-optimal Approximate Discrete and Continuous Submodular Function Minimization

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 tildeO(n/ε2) oracle evaluations with high probability. This improves over the tildeO(n5/3/ε2) oracle evaluation algorithm of Chakrabarty etal~(STOC 2017) and the tildeO(n3/2/ε2) oracle evaluation algorithm of Hamoudi etal. 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 fracpartial2fpartial xi partial xj ≤ 0 for all i neq j and is L-Lipschitz with respect to the L^∞-norm we give an algorithm that computes an ε-additive approximate minimizer with tildeO(n · poly(L/ε)) function evaluation with high probability.