List of triangles to minimum amount of convex polygons I'm in search for an algorithm that takes the left side of the following graphic as an input and outputs a minimal amount of convex polygons as seen on the right side of the graphic.



The input mesh can be very complex and could contain several holes. The position of the vertices are not constrained to a grid.

The use case for this is the creation of multiple polygon shapes for the Box2D physics engine.

Additional information:


The result does not need to be deterministic.
The result does not need to be vertically striped. 
As trichoplax noted, the right part of the picture does not resemble the minimal polygon count that can be achieved. It's just an example.
It's not necessary that it will find the absolute minimum of convex polygons. A fast algorithm is preferred over a "best solution" one.
If a edge corner is concave then it needs to border 2 of the output polygons.

So one algorithm would be to find all concave corners (including the ones in the holes) and making cuts starting from them to other concave corners. This will split the polygon in two or join 2 holes into 1 or connect a hole to the outside. 

The cut from a concave corner should be constrained by the angle or it will create a new concave corner that then needs to be cut again.