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Title: Probing to Minimize
We develop approximation algorithms for set-selection 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 well-studied. 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 set-selection 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 near-optimal solutions to these threshold problems, we obtain bicriteria adaptive policies. We apply this method to obtain an adaptive approximation algorithm for the Min-Element 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 Min-Element problem, where our objective is the sum of the smallest k element-weights, or the weight of the min-weight 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 near-optimality via coupling arguments.  more » « less
Award ID(s):
1955785 2006953 1907820 2007733
NSF-PAR ID:
10326998
Author(s) / Creator(s):
; ;
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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