A manually-checkable proof for the NP-hardness of 11-color pattern self-assembly tileset synthesis
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Patterned self-assembly tile set synthesis (pats) aims at minimizing the number of distinct DNA tile types used to self-assemble a given rectangular color pattern. For an integer k, k-pats is the subproblem of pats that restricts input patterns to those with at most k colors. We give an efficient Open image in new window verifier, and based on that, we establish a manually-checkable proof for the NP-hardness of 11-pats; the best previous manually-checkable proof is for 29-pats.
KeywordsDNA pattern self-assembly Tile complexity Manually-checkable proof
We are very thankful to the anonymous referees for their valuable comments on the earlier versions of this paper. This work is supported in part by NSF Grants CCF-1049899 and CCF-1217770 to M-Y. Kao and by HIIT Pump Priming Grant 902184/T30606, Academy of Finland, Postdoctoral Researcher Grant 13266670/T30606, JST Program to Disseminate Tenure Tracking System, MEXT, JAPAN, No. 6F36, and JSPS Grant-in-Aid for Research Activity Start-up No. 15H06212 to S. Seki.
- Cook M, Rothemund PWK, Winfree E (2004) Self-assembled circuit patterns. In: Proceedings of the 9th International Workshop on DNA Based Computers (DNA 9), LNCS, vol. 2943. Springer, p 91–107Google Scholar
- Evans CG (2014) Crystals that count! physical principles and experimental investigations of DNA tile self-assembly. Ph.D. thesis, California Institute of TechnologyGoogle Scholar
- Johnsen A, Kao MY, Seki S (2013) Computing minimum tile sets to self-assemble patterns in 29-colors. In: Proceedings of the 24th International Symposium on Algorithms and Computation (ISAAC 2013), LNCS, vol. 8283. Springer, p 699–710Google Scholar
- Kari L, Kopecki S, Étienne Meunier P, Patitz MJ, Seki S (2015a) Binary pattern tile set synthesis is NP-hard. In: Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP 2015), LNCS, vol. 9134. Springer, p 1022–1034Google Scholar
- Kari L, Kopecki S, Seki S (2015b) 3-color bounded patterned self-assembly. Nat Comput 14(2):279–292Google Scholar
- Schaefer TJ (1978) The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC 1978), p 216–226Google Scholar
- Seki S (2013) Combinatorial optimization in pattern assembly (extended abstract). In: Proceedings of the 12th International Conference on Unconventional Computation and Natural Computation (UCNC 2013), LNCS, vol. 7956. Springer, p 220–231Google Scholar
- Stefanovic D, Turberfield A (eds) (2012) The 18th International Conference on DNA Computing and Molecular Programming, Aarhus, Denmark, 14–17 August 2012Google Scholar
- Winfree E (1998) Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of TechnologyGoogle Scholar