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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Probing to Minimize
We develop approximation algorithms for setselection problems with deterministic constraints, but random objective values, i.e., stochastic probing problems. When the goal is to maximize the objective, approximation algorithms for probing problems are wellstudied. On the other hand, few techniques are known for minimizing the objective, especially in the adaptive setting, where information about the random objective is revealed during the setselection process and allowed to influence it. For minimization problems in particular, incorporating adaptivity can have a considerable effect on performance. In this work, we seek approximation algorithms that compare well to the optimal adaptive policy.
We develop new techniques for adaptive minimization, applying them to a few problems of interest. The core technique we develop here is an approximate reduction from an adaptive expectation minimization problem to a set of adaptive probability minimization problems which we call threshold problems. By providing nearoptimal solutions to these threshold problems, we obtain bicriteria adaptive policies.
We apply this method to obtain an adaptive approximation algorithm for the MinElement problem, where the goal is to adaptively pick random variables to minimize the expected minimum value seen among them, subject to a knapsack constraint. This partially resolves an open problem raised in [Goel et al., 2010]. We further consider three extensions on the MinElement problem, where our objective is the sum of the smallest k elementweights, or the weight of the minweight basis of a given matroid, or where the constraint is not given by a knapsack but by a matroid constraint. For all three of the variations we explore, we develop adaptive approximation algorithms for their corresponding threshold problems, and prove their nearoptimality via coupling arguments.
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 NSFPAR ID:
 10326998
 Editor(s):
 Braverman, Mark
 Date Published:
 Journal Name:
 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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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

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