Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Fielbaum, Andrés; Morales, Ignacio; Verschae, José https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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A Water-Filling Primal-Dual Algorithm for Approximating Non-Linear Covering Problems

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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}
}

Keywords: Knapsack-Cover Inequalities, Non-Linear Knapsack-Cover, Primal-Dual, Water-Filling Algorithm
Seminar: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Issue date: 2020
Date of publication: 29.06.2020


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