文章基本信息

标题：A Water-Filling Primal-Dual Algorithm for Approximating Non-Linear Covering Problems
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作者：Andrs Fielbaum ; Ignacio Morales ; Jos Verschae 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：168
页码：46:1-46:15
DOI：10.4230/LIPIcs.ICALP.2020.46
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：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+Îµ)-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+Îµ)-approximation algorithm in polynomial time, almost matching the 4-approximation algorithm known for the classical setting.
关键词：Knapsack-Cover Inequalities; Non-Linear Knapsack-Cover; Primal-Dual; Water-Filling Algorithm