I've solved this problem using binary search on answer and maximum matching algorithm (Kuhn algorithm) for checking if it possible to create perfect matching with this time.

I think you probably have error with checking line from point x on the bottom with point on top of some mountain corresponding to the condition - (obviously, no part of the path that he follows can be OVER the mountain - they can't fly).

According to statement:

"For every person i, he is planing to go left/right to some place (x,0) (where x is an integer).

 then climb directly to the destination along a straight line."

It means that any person can move to any integer (x, 0) and then go directly to the top of chosen mountain along a straight line. And on this line there should be no point over the surface of mountain.