Near-optimal Approximate Discrete and Continuous Submodular Function Minimization
(1909.00171)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 $\tilde{O}(n/\epsilon2)$ oracle evaluations with high probability. This improves over the $\tilde{O}(n{5/3}/\epsilon2)$ oracle evaluation algorithm of Chakrabarty \etal~(STOC 2017) and the $\tilde{O}(n{3/2}/\epsilon2)$ 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 $\frac{\partial2f}{\partial xi \partial xj} \le 0$ for all $i \neq j$ and is $L$-Lipschitz with respect to the $L\infty$-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.
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