License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.16
URN: urn:nbn:de:0030-drops-75657
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7565/
Kesselheim, Thomas ;
Tönnis, Andreas
Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints
Abstract
We study various generalizations of the secretary problem with submodular objective functions. Generally, a set of requests is revealed step-by-step to an algorithm in random order. For each request, one option has to be selected so as to maximize a monotone submodular function while ensuring feasibility. For our results, we assume that we are given an offline algorithm computing an alpha-approximation for the respective problem. This way, we separate computational limitations from the ones due to the online nature. When only focusing on the online aspect, we can assume alpha = 1.
In the submodular secretary problem, feasibility constraints are cardinality constraints, or equivalently, sets are feasible if and only if they are independent sets of a k-uniform matroid. That is, out of a randomly ordered stream of entities, one has to select a subset of size k. For this problem, we present a 0.31alpha-competitive algorithm for all k, which asymptotically reaches competitive ratio alpha/e for large k. In submodular secretary matching, one side of a bipartite graph is revealed online. Upon arrival, each node has to be matched permanently to an offline node or discarded irrevocably. We give a 0.207alpha-competitive algorithm. This also covers the problem, in which sets of entities are feasible if and only if they are independent with respect to a transversal matroid. In both cases, we improve over previously best known competitive ratios, using a generalization of the algorithm for the classic secretary problem.
Furthermore, we give an O(alpha d^(-2/(B-1)))-competitive algorithm for submodular function maximization subject to linear packing constraints. Here, d is the column sparsity, that is the maximal number of none-zero entries in a column of the constraint matrix, and B is the minimal capacity of the constraints. Notably, this bound is independent of the total number of constraints. We improve the algorithm to be O(alpha d^(-1/(B-1)))-competitive if both d and B are known to the algorithm beforehand.
BibTeX - Entry
@InProceedings{kesselheim_et_al:LIPIcs:2017:7565,
author = {Thomas Kesselheim and Andreas T{\"o}nnis},
title = {{Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
pages = {16:1--16:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-044-6},
ISSN = {1868-8969},
year = {2017},
volume = {81},
editor = {Klaus Jansen and Jos{\'e} D. P. Rolim and David Williamson and Santosh S. Vempala},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7565},
URN = {urn:nbn:de:0030-drops-75657},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.16},
annote = {Keywords: Secretary Problem, Online Algorithms, Submodular Maximization}
}
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Keywords: |
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Secretary Problem, Online Algorithms, Submodular Maximization |
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Collection: |
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017) |
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Issue Date: |
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2017 |
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Date of publication: |
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11.08.2017 |
