In this paper, we present a framework to design approximation algorithms for capacitated facility location problems with penalties/outliers using LP-rounding. Primal-dual technique, which has been particularly successful in dealing with outliers and penalties, has not been very successful in dealing with capacities. On the other hand, no primal-dual solution has been able to break the hardness of capacitated facility location problem(CFLP). LP-rounding techniques had also not been very successful in dealing with capacities until a recent work by Grover et al. \cite{GroverGKP18}. Their constant factor approximation violating the capacities by a small factor ($1 + \epsilon$) is promising while dealing with capacities.Though LP-rounding has not been very promising while dealing with penalties and outliers, we successfully apply it to deal with them along with capacities. That is, our results are obtained by rounding the solution to the natural LP once again exhibiting the power of LP-rounding technique. Solutions obtained by LP-rounding are easy to integrate with other LP-based algorithms.In this paper, we apply our framework to obtain first constant factor approximations for capacitated facility location problem with outlier (CFLPO) and capacitated $k$-facility location problem with penalty(C$k$FLPP) for hard uniform capacities using LP-rounding. Our solutions incur slight violations in capacities, ($1 + \epsilon$) for the problems without cardinality($k$) constraint and ($2 + \epsilon$) for the problems with the cardinality constraint. For the outlier variant, we also incur a small loss ($1 + \epsilon$) in outliers. Due to the hardness of the underlying problems, the violations are inevitable. Thus we achieve the best possible by rounding the solution of natural LP for these problems. To the best of our knowledge, no results are known for CFLPO and C$k$FLPP.
翻译:在本文中,我们提出了一个框架,用于设计用于使用LP四舍五入处理电离层定位问题的精密电路定位算法。 在处理外值和处罚方面特别成功的纯二元技术在处理能力方面并不十分成功。 另一方面,没有原始双向解决方案能够打破功能化设施定位问题(CFLP)的难度。 LP四舍五入技术在应对能力方面也非常不成功,直到Grover 和al. (cite{GroverGKP18})最近一项小成本工作之前,LP四舍五入电路路流定位问题。 LP四舍五入的不断要素接近能力(1+\epslon)在应对能力方面很有希望。在处理罚款和外值时,我们成功地运用它来应对能力问题。我们通过对自然液化电路流变电路流的解决方案再次展示了LP的功率。 LP四舍获得的解决方案,我们第一次在使用硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性硬性调调调 。