We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries.
We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $11/e\epsilon$ approximation for monotone functions and a $1/e\epsilon$ approximation for nonmonotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(\log^2{n}/\epsilon^3)$, which is an exponential speedup over the existing algorithms.
We obtain the first parallel algorithm for nonmonotone submodular maximization subject to packing constraints. Our algorithm achieves a $1/e\epsilon$ approximation using $O(\log(n/\epsilon) \log(1/\epsilon) \log(n+m)/ \epsilon^2)$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $11/e\epsilon$ approximation in $O(\log(n/\epsilon)\log(m)/\epsilon^2)$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective (Mahoney et al., 2016).
Our results apply more generally to the problem of maximizing a diminishing returns submodular (DRsubmodular) function.
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Submodular Maximization with Nearlyoptimal Approximation and Adaptivity in Nearlylinear Time
In this paper, we study the tradeoff between the approximation guarantee and adaptivity for the problem of maximizing a monotone submodular function subject to a cardinality constraint. The adaptivity of an algorithm is the number of sequential rounds of queries it makes to the evaluation oracle of the function, where in every round the algorithm is allowed to make polynomiallymany parallel queries. Adaptivity is an important consideration in settings where the objective function is estimated using samples and in applications where adaptivity is the main running time bottleneck. Previous algorithms achieving a nearlyoptimal $1  1/e  \epsilon$ approximation require $\Omega(n)$ rounds of adaptivity. In this work, we give the first algorithm that achieves a $1  1/e  \epsilon$ approximation using $O(\ln{n} / \epsilon^2)$ rounds of adaptivity. The number of function evaluations and additional running time of the algorithm are $O(n \; \mathrm{poly}(\log{n}, 1/\epsilon))$.
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 NSFPAR ID:
 10087828
 Date Published:
 Journal Name:
 Proceedings of the annual ACMSIAM Symposium on Discrete Algorithms
 ISSN:
 10719040
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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