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  Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints

Kesselheim, T., & Tönnis, A. (2016). Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints. Retrieved from http://arxiv.org/abs/1607.08805.

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Latex : Submodular Secretary Problems: {C}ardinality, Matching, and Linear Constraints

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 Creators:
Kesselheim, Thomas1, Author           
Tönnis, Andreas2, Author
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1Algorithms and Complexity, MPI for Informatics, Max Planck Society, ou_24019              
2External Organizations, ou_persistent22              

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Free keywords: Computer Science, Data Structures and Algorithms, cs.DS
 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. That is, out of a randomly ordered stream of entities, one has to select a subset size $k$. For this problem, we present a $0.31\alpha$-competitive algorithm for all $k$, which asymptotically reaches competitive ratio $\frac{\alpha}{e}$ 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 an $\frac{\alpha}{4}$-competitive algorithm. 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 $O(\alpha d^{-\frac{2}{B-1}})$-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 $O(\alpha d^{-\frac{1}{B-1}})$-competitive if both $d$ and $B$ are known to the algorithm beforehand.

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Language(s): eng - English
 Dates: 2016-07-292016
 Publication Status: Published online
 Pages: 22 p.
 Publishing info: -
 Table of Contents: -
 Rev. Type: -
 Identifiers: arXiv: 1607.08805
URI: http://arxiv.org/abs/1607.08805
BibTex Citekey: DBLP:journals/corr/KesselheimT16a
 Degree: -

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