Interior point methods that follow the primal-dual central path of a dual pair of linear programs (Po),(Do) require that these problems are strictly feasible. To get around this difficulty, one technique is to embed (Po),(Do) into a family of suitably perturbed strictly feasible linear programs (Pr),(Dr),r>0and to follow the path (x(r),s(r)),r>0, of all strictly feasible solutions of (Pr),(Dr) withxi(r)si(r)=r,i= 1…,n. Path following methods compute approximations to this path at parametersR=ro>r1>…, and their complexity is measured by the numberN=N(r, R) of steps needed to reduce a givenR>0 to a desiredr>0. It is shown for a standard method thatN(r, R) is estimated by a boundwhereCois a universal constant but K=K(ƀ,ȼ)is a constant depending ƀ,ȼ on with K(0,0)=0, and on the sizeof the path.