Flow thinning (FT) is a traffic protection mechanism for communication networks with variable link capacities, for example wireless networks. With FT, end-to-end traffic demands are equipped with dedicated logical tunnels, for example MPLS tunnels, whose nominal capacity is subject to thinning in order to follow the fluctuations of link capacities availability. Moreover, instantaneous traffic of each demand is throttled at its originating node accordingly to the current total capacity available on the demand's dedicated tunnels. In effect, the network is always capable of carrying the admitted traffic. In this paper we deal with an efficient, implementable version of FT, referred to as AFT (affine flow thinning). By deriving appropriate cutting-plane and path-generation algorithms, we show how real-life network dimensioning problems for AFT can be efficiently treated using a proper characterization of the network link availability states. Results of a numerical study illustrate tractability of the cost minimization problems, and assess cost efficiency of AFT in comparison with other protection mechanisms.