Attenuating transverse isotropy with a vertical symmetry axis (VTI) can be used to determine the directional variation of the wave attenuation in finely layered structures. Specially, complex-valued traveltimes can be used in absorption compensation, imaging and Q tomography etc. The acoustic attenuating VTI eikonal equation governs the complex-valued traveltimes of P-waves in such a medium, whereas the real and imaginary parts of the traveltimes describes the wave phase behavior and its energy absorption, respectively. We use perturbation theory to design two fast sweeping algorithms for solving the acoustic attenuating VTI eikonal equation. Through numerical tests, we study the accuracy and robustness of these algorithms. We find that the algorithm corresponding to the perturbation formulation using only the attenuation parameters is more robust and provides a stable solution compared to the algorithm developed by perturbing both anellipticity and anisotropy parameters. The lessons learned here are vital in the effort to develop a stable algorithm for eikonal equations corresponding to attenuating anisotropic media.