  We study the problem of decomposing (i.e. partitioning and covering) polygons
into components that are $\alpha$-fat, which means that the aspect ratio of
each subpolygon is at most $\alpha$. We consider decompositions without Steiner
points. We present a polynomial-time algorithm for simple polygons that finds
the minimum $\alpha$ such that an $\alpha$-fat partition exists. Furthermore,
we show that finding an $\alpha$-fat partition or covering with minimum
cardinality is NP-hard for polygons with holes.
