Before revision:  However , we use a  more general class of polygons -- polygons that are horizontally or vertically convex -- which allows us to provide an arguably simpler analysis and already improve the approximation ratio .
After revision:  However , we use a more general class of polygons -- polygons that are horizontally or vertically convex -- which allows us to provide an arguably simpler analysis and at the same time already improves the approximation ratio .