Near-optimal Approximate Discrete and Continuous Submodular Function Minimization

Abstract: 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 $\tilde{O}(n/\epsilon<sup>2)$ oracle evaluations with high probability. This improves over the $\tilde{O}(n<sup>{5/3}/\epsilon<sup>2)$ oracle evaluation algorithm of Chakrabarty \etal~(STOC 2017) and the $\tilde{O}(n<sup>{3/2}/\epsilon<sup>2)$ 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 $[0, 1]<sup>n$ that satisfies $$\frac{\partial^2f}{\partial x_i \partial x_j} \le 0$$ for all $i \neq j$ and is $L$-Lipschitz with respect to the $L<sup>\infty$-norm we give an algorithm that computes an $\epsilon$-additive approximate minimizer with $\tilde{O}(n \cdot \mathrm{poly}(L/\epsilon))$ function evaluation with high probability.