3-color bounded patterned self-assembly

The problem of patterned self-assembly tile set synthesis (Pats) is to find a minimal tile set which uniquely self-assembles into a given pattern. Czeizler and Popa proved the NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {NP}$$\end{document}-completeness of Pats and Seki showed that the Pats problem is already NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {NP}$$\end{document}-complete for patterns with 60 colors. In search for the minimal number of colors such that Pats remains NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {NP}$$\end{document}-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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {NP}$$\end{document}-complete for patterns with just three colors and, as a byproduct of this result, we also obtain a novel proof for the NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {NP}$$\end{document}-completeness of Pats which is more concise than the previous proofs.