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.MFCS.2018.31
URN: urn:nbn:de:0030-drops-96138
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9613/
Cellinese, Francesco ;
D'Angelo, Gianlorenzo ;
Monaco, Gianpiero ;
Velaj, Yllka
Generalized Budgeted Submodular Set Function Maximization
Abstract
In this paper we consider a generalization of the well-known budgeted maximum coverage problem. We are given a ground set of elements and a set of bins. The goal is to find a subset of elements along with an associated set of bins, such that the overall cost is at most a given budget, and the profit is maximized. Each bin has its own cost and the cost of each element depends on its associated bin. The profit is measured by a monotone submodular function over the elements.
We first present an algorithm that guarantees an approximation factor of 1/2(1-1/e^alpha), where alpha <= 1 is the approximation factor of an algorithm for a sub-problem. We give two polynomial-time algorithms to solve this sub-problem. The first one gives us alpha=1- epsilon if the costs satisfies a specific condition, which is fulfilled in several relevant cases, including the unitary costs case and the problem of maximizing a monotone submodular function under a knapsack constraint. The second one guarantees alpha=1-1/e-epsilon for the general case. The gap between our approximation guarantees and the known inapproximability bounds is 1/2.
We extend our algorithm to a bi-criterion approximation algorithm in which we are allowed to spend an extra budget up to a factor beta >= 1 to guarantee a 1/2(1-1/e^(alpha beta))-approximation. If we set beta=1/(alpha)ln (1/(2 epsilon)), the algorithm achieves an approximation factor of 1/2-epsilon, for any arbitrarily small epsilon>0.
BibTeX - Entry
@InProceedings{cellinese_et_al:LIPIcs:2018:9613,
author = {Francesco Cellinese and Gianlorenzo D'Angelo and Gianpiero Monaco and Yllka Velaj},
title = {{Generalized Budgeted Submodular Set Function Maximization}},
booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
pages = {31:1--31:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-086-6},
ISSN = {1868-8969},
year = {2018},
volume = {117},
editor = {Igor Potapov and Paul Spirakis and James Worrell},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9613},
URN = {urn:nbn:de:0030-drops-96138},
doi = {10.4230/LIPIcs.MFCS.2018.31},
annote = {Keywords: Submodular set function, Approximation algorithms, Budgeted Maximum Coverage}
}
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Keywords: |
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Submodular set function, Approximation algorithms, Budgeted Maximum Coverage |
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Collection: |
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43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018) |
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Issue Date: |
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2018 |
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Date of publication: |
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27.08.2018 |
