2 research outputs found
Submodular Maximization with Matroid and Packing Constraints in Parallel
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
approximation for monotone functions and a
approximation for non-monotone functions, which nearly matches the best
guarantees known in the fully adaptive setting. The number of rounds of
adaptivity is , which is an exponential speedup over
the existing algorithms.
We obtain the first parallel algorithm for non-monotone submodular
maximization subject to packing constraints. Our algorithm achieves a
approximation using parallel rounds, which is again an exponential speedup
in parallel time over the existing algorithms. For monotone functions, we
obtain a approximation in
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.
Our results apply more generally to the problem of maximizing a diminishing
returns submodular (DR-submodular) function
Unconstrained Submodular Maximization with Constant Adaptive Complexity
In this paper, we consider the unconstrained submodular maximization problem.
We propose the first algorithm for this problem that achieves a tight
-approximation guarantee using
adaptive rounds and a linear number of function evaluations. No previously
known algorithm for this problem achieves an approximation ratio better than
using less than rounds of adaptivity, where is the size
of the ground set. Moreover, our algorithm easily extends to the maximization
of a non-negative continuous DR-submodular function subject to a box constraint
and achieves a tight -approximation guarantee for this
problem while keeping the same adaptive and query complexities.Comment: Authors are listed in alphabetical orde