Combinatorial Optimization in Pattern Assembly

TL;DR

This paper studies the complexity of pattern self-assembly tile set synthesis (PATS), proving NP-hardness for certain cases and analyzing properties of minimal solutions for simple patterns.

Contribution

It introduces a specific 59-colored pattern and demonstrates NP-hardness of 59-PATS by reduction from 3SAT, advancing understanding of PATS complexity.

Findings

NP-hardness of 59-PATS established

Properties of minimal RTASs for simple patterns analyzed

Reduction from 3SAT to 59-PATS demonstrated

Abstract

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 first investigate simple patterns and properties of minimum RTASs for them. Then based on them, we design a 59-colored pattern to which 3SAT is reduced, and prove that 59-PATS is NP-hard.