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

Authors Andrés Fielbaum, Ignacio Morales, José Verschae



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Andrés Fielbaum
  • Department of Cognitive Robotics, Faculty of Mechanical, Maritime and Materials Engineering, TU Delft, The Netherlands
Ignacio Morales
  • Departamento de Ingeniería Industrial, Escuela de Ingeniería, Pontificia Universidad Católica, Santiago, Chile
José Verschae
  • Instituto de Ingeniería Matemática y Computacional, Facultad de Matemáticas y Escuela de Ingeniería, Pontificia Universidad Católica, Santiago, Chile

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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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