This paper concerns the Lateral Gene Transfer Problem. This minimization problem, defined by Hallet and Lagergren [6], is that of finding the most parsimonious lateral gene transfer scenario for a given pair of gene and species trees. Our main results are the following: (a) We show that it is not possible to approximate the problem in polynomial time within an approximation ratio of 1+ε, for some constant ε > 0 unless P=NP. We also provide explicit values of ε for the above claim. (b) We provide an upper bound on the cost of any 1-active scenario and prove the tightness of this bound.