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 epsilon-additive approximate minimizer in (O) over tilde (n="2) oracle evaluations with high probability. This improves over the (O) over tilde (n(5/3)/epsilon(2)) oracle evaluation algorithm of Chakrabarty et al. (STOC 2017) and the (O) over tilde (n(3/2)/epsilon(2)) oracle evaluation algorithm of Hamoudi et al.. 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 partial derivative(2)f/partial derivative x(i)partial derivative x(j) <= 0 for all i not equal j and is L-Lipschitz with respect to the L-infinity-norm we give an algorithm that computes an epsilon-additive approximate minimizer with (O) over tilde (n.poly(L/epsilon)) function evaluation with high probability.