Optimal transport based misfit functions have been introduced to mitigate cycle skipping in full waveform inversion. Recently, we have proposed to compare the discrete graph of the data through optimal transport. This strategy makes possible to interpret non-positive data through optimal transport (OT), while maintaining its convexity property with respect to time and amplitude shifts. Here, we introduce a novel numerical strategy, which makes the computational overcost associated with the graph space OT approach almost negligible compared to a conventional least-squares distance. This is possible through the interpretation of the graph space OT problem as a linear sum assignment problem, for which specific algorithms exist: we select the auction algorithm for its good performance on small scale, dense instances of such problems. We introduce a tuning parameter, based on the estimation of the maximum time shift, to scale the misfit function properly. Application on visco-acoustic synthetic data illustrate the interest of the strategy. From significantly cycle-skipped data, the method retrieves a meaningful estimation of the velocity model, while the conventional least-squares approach and one previously introduced OT based approach both fail. The computational cost increase is only few percents compared to the least-squares misfit function.