Facility location problem is a well established research area within Operations Research. Capacitated facility location problem is one of the most important variants, in which each facility has an upper bound on the demand, i.e., capacity. Moreover, the integrality gap of the natural linear program relaxation for the problem is infinite. Fortunately, the gap is finite when all opening costs for the facilities are the same. We consider a capacity facility location problem with penalties in which it is allowed some customers are not served by facilities and all opening costs are uniform. Based on LP-rounding framework, we propose a 5.732-approximation algorithm.

设施位置问题是运筹学中一个完善的研究领域。受限制的设施选址问题是最重要的变体之一，其中每个设施的需求（即容量）都有上限。而且，自然线性程序松弛对于该问题的积分间隙是无限的。幸运的是，当所有设施的开放成本都相同时，差距是有限的。我们认为有能力的设施的位置问题会受到惩罚，其中允许某些客户无法获得设施服务，并且所有开业费用都是统一的。基于LP舍入框架，我们提出了5.732近似算法。