  In this paper, we study how to fold a specified origami crease pattern in
order to minimize the impact of paper thickness. Specifically, origami designs
are often expressed by a mountain-valley pattern (plane graph of creases with
relative fold orientations), but in general this specification is consistent
with exponentially many possible folded states. We analyze the complexity of
finding the best consistent folded state according to two metrics: minimizing
the total number of layers in the folded state (so that a "flat folding" is
indeed close to flat), and minimizing the total amount of paper required to
execute the folding (where "thicker" creases consume more paper). We prove both
problems strongly NP-complete even for 1D folding. On the other hand, we prove
the first problem fixed-parameter tractable in 1D with respect to the number of
layers.
