We analyze the complexity of two fundamental verification problems within a generalization of the two-handed tile self-assembly model (2HAM) where initial system assemblies are not restricted to be singleton tiles, but may be larger pre-built assemblies. Within this model we consider the producibility problem, which asks if a given tile system builds, or produces, a given assembly, and the unique assembly verification (UAV) problem, which asks if a given system uniquely produces a given assembly. We show that producibility is NP-complete and UAV is coNP^{NP}-complete even when the initial assembly size and temperature threshold are both bounded by a constant. This is in stark contrast to results in the standard model with singleton input tiles where producibility is in P and UAV is in coNP for 𝒪(1) bounded temperature and coNP-complete when temperature is part of the input. We further provide preliminary results for producibility and UAV in the case of 1-dimensional linear assemblies with pre-built assemblies, and provide polynomial time solutions.
@InProceedings{caballero_et_al:LIPIcs.SAND.2022.8, author = {Caballero, David and Gomez, Timothy and Schweller, Robert and Wylie, Tim}, title = {{Complexity of Verification in Self-Assembly with Prebuilt Assemblies}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.8}, URN = {urn:nbn:de:0030-drops-159503}, doi = {10.4230/LIPIcs.SAND.2022.8}, annote = {Keywords: 2-handed assembly, verification, prebuilt} }
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