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# A Water-Filling Primal-Dual Algorithm for Approximating Non-Linear Covering Problems

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LIPIcs.ICALP.2020.46.pdf
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## Cite As

Andrés Fielbaum, Ignacio Morales, and José Verschae. A Water-Filling Primal-Dual Algorithm for Approximating Non-Linear Covering Problems. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.46

## 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Packing and covering problems
• Mathematics of computing → Discrete optimization
• Mathematics of computing → Linear programming
##### Keywords
• Knapsack-Cover Inequalities
• Non-Linear Knapsack-Cover
• Primal-Dual
• Water-Filling Algorithm

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