I have the same problem. Grade on my front lawn rises about three feet from the sidewalk at about a 30° slope, so there's a pretty steep little hill upon which most of the snow from my shovelling ends up. The depth of the snow and ice on the street is higher than my sidewalk too, so there really isn't anywhere for the water to go as the snow melts.
I salt the #### out of it and hope for the best.