Combinatorial optimization problems in self-assembly

Abstract

Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum size, and the second to finding optimum running time for a program to assemble the shape.Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum size, and the second to finding optimum running time for a program to assemble the shape.We prove that the first problem is NP-complete in general, and polynomial time solvable on trees and squares. In order to prove that the problem is in NP, we present a polynomial time algorithm to verify whether a given tile system uniquely produces a given shape. This algorithm is analogous to a verifier for traditional computational systems, and may well be of independent interest. For the second problem, we present a polynomial time $O(\log n)$-approximation algorithm that works for a large class of tile systems that we call partial order systems.