  Patterned self-assembly tile set synthesis PATS is the problem of finding a
minimal tile set which uniquely self-assembles into a given pattern. Czeizler
and Popa proved the NP-completeness of PATS and Seki showed that the PATS
problem is already NP-complete for patterns with 60 colors. In search for the
minimal number of colors such that PATS remains NP-complete, we introduce
multiple bound PATS (mbPATS) where we allow bounds for the numbers of tile
types of each color. We show that mbPATS is NP-complete for patterns with just
three colors and, as a byproduct of this result, we also obtain a novel proof
for the NP-completeness of PATS which is more concise than the previous proofs.
