In this paper we consider the problem of computing control invariant sets for linear controlled systems with constraints on the input and on the states. We focus in particular on the complexity of the computation of the N-step operator, given by the Minkowski addition of sets, that is the basis of many of the iterative procedures for obtaining control invariant sets. Set inclusions conditions for control invariance are presented that involve the N-step sets and are posed in form of linear programming problems. Such conditions are employed in algorithms based on LP problems that allow to overcome the complexity limitation inherent to the set addition and can be applied also to high dimensional systems. The efficiency and scalability of the method are illustrated by computing in less than two seconds an approximation of the maximal control invariant set, based on the 15-step operator, for a system whose state and input dimensions are 20 and 10 respectively.