TY - GEN

T1 - A water-filling primal-dual algorithm for approximating non-linear covering problems

AU - Fielbaum, Andrés

AU - Morales, Ignacio

AU - Verschae, José

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Knapsack-Cover Inequalities

KW - Non-Linear Knapsack-Cover

KW - Primal-Dual

KW - Water-Filling Algorithm

UR - http://www.scopus.com/inward/record.url?scp=85089346460&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ICALP.2020.46

DO - 10.4230/LIPIcs.ICALP.2020.46

M3 - Conference contribution

AN - SCOPUS:85089346460

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - Proceedings 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020

A2 - Czumaj, Artur

A2 - Dawar, Anuj

A2 - Merelli, Emanuela

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

CY - Dagstuhl, Germany

T2 - 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020

Y2 - 8 July 2020 through 11 July 2020

ER -