Computing Minimum Tile Sets to Self-Assemble Color Patterns
نویسندگان
چکیده
Patterned self-assembly tile set synthesis (PATS) aims at finding a minimum tile set to uniquely self-assemble a given rectangular color pattern. For k ≥ 1, k-PATS is a variant of PATS that restricts input patterns to those with at most k colors. We prove the NP-hardness of 29-PATS, where the best known is that of 60-PATS.
منابع مشابه
A manually-checkable proof for the NP-hardness of 11-color pattern self-assembly tile set synthesis
Patterned self-assembly tile set synthesis (Pats) aims at finding a minimum tile set to uniquely self-assemble a given rectangular (color) pattern. For k ≥ 1, k-Pats is a variant of Pats that restricts input patterns to those with at most k colors. A computer-assisted proof has been recently proposed for 2-Pats by Kari et al. [arXiv:1404.0967 (2014)]. In contrast, the best known manually-checka...
متن کاملSynthesizing Minimal Tile Sets for Patterned DNA Self-assembly
The Pattern self-Assembly Tile set Synthesis (PATS) problem is to determine a set of coloured tiles that self-assemble to implement a given rectangular colour pattern. We give an exhaustive branch-and-bound algorithm to find tile sets of minimum cardinality for the PATS problem. Our algorithm makes use of a search tree in the lattice of partitions of the ambient rectangular grid, and an efficie...
متن کاملComplexities for High-Temperature Two-Handed Tile Self-assembly
Tile self-assembly is a formal model of computation capturing DNA-based nanoscale systems. Here we consider the popular twohanded tile self-assembly model or 2HAM. Each 2HAM system includes a temperature parameter, which determines the threshold of bonding strength required for two assemblies to attach. Unlike most prior study, we consider general temperatures not limited to small, constant val...
متن کاملSelf-Assembly of Infinite Structures
We review some recent results related to the self-assembly of infinite structures in the Tile Assembly Model. These results include impossibility results, as well as novel tile assembly systems in which shapes and patterns that represent various notions of computation self-assemble. Several open questions are also presented and motivated.
متن کاملCombinatorial Optimization in Pattern Assembly
Pattern self-assembly tile set synthesis (Pats) is a combinatorial optimization problem which aim at minimizing a rectilinear tile assembly system (RTAS) that uniquely self-assembles a given rectangular pattern, and is known to be NP-hard. Pats gets practically meaningful when it is parameterized by a constant c such that any given pattern is guaranteed to contain at most c colors (c-Pats). We ...
متن کامل