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 . Using a new fractional charging argument and fork-fences to guide the partitions , we improve the approximation ratio even more to 4.
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 already improve the approximation ratio to 6. Then, using a more elaborate charging scheme and a recursive partitioning into general axis-parallel polygons with constant complexity , we improve the approximation ratio even more to 4.