We present a novel approach to the computation of dense correspondence maps between shapes in a non-rigid setting. The problem is defined in terms of functional correspondences. We deal with the non-injectivity of the solution of the functional map framework due to the under-determinedness of the original problem. Key to our approach is the injectivity constraint plugged directly into the problem to optimize, achieved casting it as an assignment problem. This leads to an iterative process which yields a high quality bijective map between the shapes. In the experimental section we present both quantitative and qualitative results, showing that the proposed approach is competitive with the current state-of-the-art on quasi-isometric shape matching benchmarks.