0.4633-Approximation for Online and Offline on-Monotone DR-Submodular Maximization
Vaneet Aggarwal, Donglei Du
We obtain the approximation coefficient $α_0=54440750000/117486474169>0.463378$ for maximizing nonnegative, non-monotone DR-submodular functions over compact convex down-closed subsets of the unit cube, with queries allowed throughout the cube. This improves the previous best coefficient, $4/9$. A coordinatewise map, exponential on small coordinates and linear-fractional on large ones, turns the learner's state, a feasible point, into a feasible action whose value is at least $α_0$ times that of any feasible point, up to a linear term. The proof is analytical and reduces to one-dimensional inequalities. The linear term has an unbiased estimator from one noisy-gradient query, so online maximization reduces to linear learning. Under a uniform bound on the gradients, the expected approximate regret is $O(\sqrt T)$ with projected ascent and $O(T^{3/4})$ with one linear-maximization call per round; with one noisy value per round, it is $O(T^{3/4})$ and $O(T^{7/8})$. Online-to-batch conversion gives offline guarantees. With exact values and gradients, contracting the objective toward a feasible interior point yields an expected $(α_0-\varepsilon)$-approximation for every nonnegative $C^1$ DR-submodular objective, with oracle counts polynomial in $n$ and $\varepsilon^{-1}$ and no gradient or smoothness parameter. Sampling and standard rounding give $(α_0-\varepsilon)$-approximation for submodular set functions under one matroid or a fixed number of knapsack constraints.
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