Submodular Maximization with Packing Constraints in Parallel. (arXiv:1808.09987v1 [cs.DS])

We consider the problem of maximizing the multilinear extension of a submodular function subject to packing constraints in parallel. For monotone functions, we obtain a $1-1/e-\epsilon$ approximation using $O(\log(n/\epsilon)\log(m)/\epsilon^2)$ rounds of adaptivity and evaluations of the function and its gradient, where $m$ is the number of packing constraints and $n$ is the number of variables. For non-monotone functions, we obtain a $1/e-\epsilon$ approximation using $O(\log(n/\epsilon)\log(1/\epsilon)\log(n+m)/\epsilon^2)$ rounds of adaptivity and evaluations of the function and its gradient. Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function subject to packing constraints.

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