This work addresses the facility location problem, a key area of research in Operations Research and Artificial Intelligence. Specifically, we examine a competitive facility location problem where a firm seeks to establish new facilities in a market already served by existing competitors. To predict customer demand, we utilize a general class of customer behavior models, known as the nested-logit model, which is widely recognized as one of the most popular demand models in the literature. The facility location problem under the nested-logit model is characterized by its high nonlinearity and complexity. Existing methods either do not operate within polynomial time or fail to guarantee near-optimal solutions at any desired level of precision. In this study, by leveraging the unique structure of the nested logit choice model, we propose a Fully Polynomial-Time Approximation Scheme (FPTAS) to efficiently solve the problem. To the best of our knowledge, this is the first FPTAS developed for this type of competitive facility location problem.