 We show that the assortment optimization problem under this model is NP-hard and present a fully polynomial-time approximation scheme (FPTAS) for this problem .
<meaning-changed>  We show that the assortment optimization problem under this model is NP-hard and present a fully polynomial-time approximation scheme (FPTAS) for this problem .
 We show that the assortment optimization problem under this model is NP-hard and present a fully polynomial-time approximation scheme (FPTAS) under reasonable assumptions .