This paper deals with computational aspects of characterization and construction of polyhedral lambda-contractive sets with respect to discrete-time linear systems with saturating feedback control inputs. Using a piecewise-affine model of the saturating closed-loop system, new necessary and sufficient algebraic condition for convex closed polyhedra be lambda-contractive is derived. Based on linear programming formulation of this condition, an effective procedure is proposed for construction of as large as possible lambda-contractive convex polyhedra for estimation of the region of asymptotic stability of origin. The procedure starts with a lambda-contractive polyhedron, possibly contained in the region of linear control, and progressively expands it non-homothetically over the region of non-linear saturated control. The proposed approach is less conservative and computationally much more efficient than previously published ones.