In this paper, we study the approximate biprojectivity and the approximate biflatness of a Banach algebra A and find some relations between theses concepts with ϕ-amenability and ϕ-contractibility, where ϕ is a character on A. Among other things, we show that θ-Lau product algebra L 1 (G) × θ A(G) is approximately biprojective if and only if G is finite, where L 1 (G) and A(G) are the group algebra and the Fourier algebra of a locally compact group G, respectively. We also characterize approximately biprojective and approximately biflat semigroup algebras associated with the inverse semigroups.