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.ICALP.2020.46
URN: urn:nbn:de:0030-drops-124531
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12453/
Fielbaum, Andrés ;
Morales, Ignacio ;
Verschae, José
A Water-Filling Primal-Dual Algorithm for Approximating Non-Linear Covering Problems
Abstract
Obtaining strong linear relaxations of capacitated covering problems constitute a significant technical challenge even for simple settings. For one of the most basic cases, the Knapsack-Cover (Min-Knapsack) problem, the relaxation based on knapsack-cover inequalities has an integrality gap of 2. These inequalities are exploited in more general problems, many of which admit primal-dual approximation algorithms.
Inspired by problems from power and transport systems, we introduce a general setting in which items can be taken fractionally to cover a given demand. The cost incurred by an item is given by an arbitrary non-decreasing function of the chosen fraction. We generalize the knapsack-cover inequalities to this setting an use them to obtain a (2+ε)-approximate primal-dual algorithm. Our procedure has a natural interpretation as a bucket-filling algorithm which effectively overcomes the difficulties implied by having different slopes in the cost functions. More precisely, when some superior segment of an item presents a low slope, it helps to increase the priority of inferior segments. We also present a rounding algorithm with an approximation guarantee of 2.
We generalize our algorithm to the Unsplittable Flow-Cover problem on a line, also for the setting of fractional items with non-linear costs. For this problem we obtain a (4+ε)-approximation algorithm in polynomial time, almost matching the 4-approximation algorithm known for the classical setting.
BibTeX - Entry
@InProceedings{fielbaum_et_al:LIPIcs:2020:12453,
author = {Andr{\'e}s Fielbaum and Ignacio Morales and Jos{\'e} Verschae},
title = {{A Water-Filling Primal-Dual Algorithm for Approximating Non-Linear Covering Problems}},
booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
pages = {46:1--46:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-138-2},
ISSN = {1868-8969},
year = {2020},
volume = {168},
editor = {Artur Czumaj and Anuj Dawar and Emanuela Merelli},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12453},
URN = {urn:nbn:de:0030-drops-124531},
doi = {10.4230/LIPIcs.ICALP.2020.46},
annote = {Keywords: Knapsack-Cover Inequalities, Non-Linear Knapsack-Cover, Primal-Dual, Water-Filling Algorithm}
}
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Keywords: |
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Knapsack-Cover Inequalities, Non-Linear Knapsack-Cover, Primal-Dual, Water-Filling Algorithm |
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Collection: |
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47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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29.06.2020 |
