The network choice revenue management problem models customers as choosing from an offerset, and the firm decides the best subset to offer at any given moment to maximize expected revenue. The resulting dynamic program for the firm is intractable and approximated by a deterministic linear program called the CDLP which has an exponential number of columns. However, under the choice-set paradigm when the segment consideration sets overlap, the CDLP is difficult to solve. Column generation has been proposed but finding an entering column has been shown to be NP-hard. In this paper, starting with a concave program formulation based on segment-level consideration sets called SDCP, we add a class of valid inequalities called product cuts, that project onto subsets of intersections. In addition we propose a natural direct tightening of the SDCP called kSDCP, and compare the performance of both methods on the benchmark data sets in the literature. Both the product cuts and the kSDCP method are very simple and easy to implement, work with general discrete choice models and are applicable to the case of overlapping segment consideration sets. In our computational testing SDCP with product cuts achieves the CDLP value at a fraction of the CPU time taken by column generation and hence has the potential to be scalable to industrial-size problems.