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A manually-checkable proof for the NP-hardness of 11-color pattern self-assembly tileset synthesis

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Abstract

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 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.

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Notes

  1. In this paper, by pattern, we always mean a rectangular color pattern.

  2. Unlike the original definition in Ma and Lombardi (2008), a solution to pats is required to be directed here, but this difference does not change the problem as a minimum RTAS is always directed (Göös et al. 2014).

  3. In contrast, for example, \((\mathtt{T}, \mathtt{F}, \mathtt{T}, \mathtt{F})\) does not satisfy \(\phi \) in the sense because it satisfies more than one literal of the first clause of \(\phi \).

  4. n does not denote a negated variable. Recall that monotonicity requires that variables are never negated in clauses.

  5. Recall that m is the number of variables involved in \(\phi \).

  6. If the second choice in Lemma 2 is made, then there are only 2 tile types left uncolored at this point.

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Acknowledgments

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.

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Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, 60208, USA

    Aleck Johnsen & Ming-Yang Kao

  2. Helsinki Institute for Information Technology (HIIT) and Department of Computer Science, Aalto University, P. O. Box 15400, 00076, Aalto, Finland

    Shinnosuke Seki

  3. Department of Communication Engineering and Informatics, The University of Electro-Communications, 1-5-1, Chofugaoka, Chofu, 1828585, Tokyo, Japan

    Shinnosuke Seki

Authors
  1. Aleck Johnsen
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  2. Ming-Yang Kao
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  3. Shinnosuke Seki
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Correspondence to Shinnosuke Seki.

Additional information

This is an improved full version of our previous works presented at conferences Johnsen et al. (2013), Seki (2013).

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Johnsen, A., Kao, MY. & Seki, S. A manually-checkable proof for the NP-hardness of 11-color pattern self-assembly tileset synthesis. J Comb Optim 33, 496–529 (2017). https://doi.org/10.1007/s10878-015-9975-6

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  • DOI: https://doi.org/10.1007/s10878-015-9975-6

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