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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},