We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a general matroid constraint with a 1  1/e  epsilon approximation that achieves a fast running time provided we have a fast data structure for maintaining an approximately maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the first algorithms for these classes of matroids that achieve a nearlyoptimal, 1  1/e  epsilon approximation, using a nearlylinear number of function evaluations and arithmetic operations.
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Towards Nearlylinear Time Algorithms for Submodular Maximization with a Matroid Constraint
We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a general matroid constraint with a $1  1/e  \epsilon$ approximation that achieves a fast running time provided we have a fast data structure for maintaining an approximately maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the first algorithms for these classes of matroids that achieve a nearlyoptimal, $1  1/e  \epsilon$ approximation, using a nearlylinear number of function evaluations and arithmetic operations.
more »
« less
 NSFPAR ID:
 10105032
 Date Published:
 Journal Name:
 International Colloquium on Automata, Languages, and Programming
 Volume:
 132
 Page Range / eLocation ID:
 54:154:14
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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