文章基本信息

标题：Lagrangian Relaxation and Partial Cover (Extended Abstract)
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作者：Juli{\'a}n Mestre
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2008
卷号：1
页码：539-550
DOI：10.4230/LIPIcs.STACS.2008.1315
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving (LMP) $alpha$-approximation as a black box can yield an approximation factor better than~$frac{4}{3} alpha$. This matches the upper bound given by K"onemann et al. (ESA 2006, pages 468--479). Faced with this limitation we study a specific, yet broad class of covering problems: Partial Totally Balanced Cover. By carefully analyzing the inner workings of the LMP algorithm we are able to give an almost tight characterization of the integrality gap of the standard linear relaxation of the problem. As a consequence we obtain improved approximations for the Partial version of Multicut and Path Hitting on Trees, Rectangle Stabbing, and Set Cover with $ ho$-Blocks.
关键词：Lagrangian Relaxation; Partial Cover; Primal-Dual Algorithms