### Abstract

The Pattern self-Assembly Tile set Synthesis (PATS) problem asks to determine a set of coloured tiles which, left alone in the solution, would self-assemble to implement a given rectangular colour pattern. Ma and Lombardi (2009) introduce and study the PATS problem from a combinatorial optimization point of view, trying to find algorithms which would minimize the required number of distinct tile types. In particular, they claimed that the above optimization problem is NP-hard. However, their NP-hardness proof turns out to be incorrect. Our main result is to give a correct NP-hardness proof via a reduction from the 3SAT. By definition, the PATS problem assumes that the assembly of a pattern starts always from an "L"-shaped seed structure, fixing the borders of the pattern. In this context, we study the assembly complexity of various pattern families and we show how to construct families of patterns which require a non-constant number of tiles to be assembled.

Original language | English |
---|---|

Pages (from-to) | 23-37 |

Number of pages | 15 |

Journal | Theoretical Computer Science |

Volume | 499 |

DOIs | |

Publication status | Published - Aug 12 2013 |

Externally published | Yes |

### Fingerprint

### Keywords

- DNA self-assembly
- Minimal tile sets
- NP-hardness
- Pattern assembly
- Tile assembly model

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*499*, 23-37. https://doi.org/10.1016/j.tcs.2013.05.009

**Synthesizing minimal tile sets for complex patterns in the framework of patterned DNA self-assembly.** / Czeizler, Eugen; Popa, Alexandru.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 499, pp. 23-37. https://doi.org/10.1016/j.tcs.2013.05.009

}

TY - JOUR

T1 - Synthesizing minimal tile sets for complex patterns in the framework of patterned DNA self-assembly

AU - Czeizler, Eugen

AU - Popa, Alexandru

PY - 2013/8/12

Y1 - 2013/8/12

N2 - The Pattern self-Assembly Tile set Synthesis (PATS) problem asks to determine a set of coloured tiles which, left alone in the solution, would self-assemble to implement a given rectangular colour pattern. Ma and Lombardi (2009) introduce and study the PATS problem from a combinatorial optimization point of view, trying to find algorithms which would minimize the required number of distinct tile types. In particular, they claimed that the above optimization problem is NP-hard. However, their NP-hardness proof turns out to be incorrect. Our main result is to give a correct NP-hardness proof via a reduction from the 3SAT. By definition, the PATS problem assumes that the assembly of a pattern starts always from an "L"-shaped seed structure, fixing the borders of the pattern. In this context, we study the assembly complexity of various pattern families and we show how to construct families of patterns which require a non-constant number of tiles to be assembled.

AB - The Pattern self-Assembly Tile set Synthesis (PATS) problem asks to determine a set of coloured tiles which, left alone in the solution, would self-assemble to implement a given rectangular colour pattern. Ma and Lombardi (2009) introduce and study the PATS problem from a combinatorial optimization point of view, trying to find algorithms which would minimize the required number of distinct tile types. In particular, they claimed that the above optimization problem is NP-hard. However, their NP-hardness proof turns out to be incorrect. Our main result is to give a correct NP-hardness proof via a reduction from the 3SAT. By definition, the PATS problem assumes that the assembly of a pattern starts always from an "L"-shaped seed structure, fixing the borders of the pattern. In this context, we study the assembly complexity of various pattern families and we show how to construct families of patterns which require a non-constant number of tiles to be assembled.

KW - DNA self-assembly

KW - Minimal tile sets

KW - NP-hardness

KW - Pattern assembly

KW - Tile assembly model

UR - http://www.scopus.com/inward/record.url?scp=84881186834&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84881186834&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2013.05.009

DO - 10.1016/j.tcs.2013.05.009

M3 - Article

VL - 499

SP - 23

EP - 37

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -