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

TL;DR

This paper presents a manually-checkable proof demonstrating that the problem of 11-color pattern self-assembly tile set synthesis (11-PATS) is NP-hard, improving upon previous proofs for higher color counts.

Contribution

The paper provides the first manually-checkable NP-hardness proof for 11-PATS, reducing the previously known NP-hardness threshold from 29 colors.

Findings

Proof of NP-hardness for 11-PATS

Reduction from known NP-hard problems

Improved bounds on pattern complexity

Abstract

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-checkable proof is for the NP-hardness of 29-PATS by Johnsen, Kao, and Seki [ISAAC 2013, LNCS 8283, pp.~699-710]. We propose a manually-checkable proof for the NP-hardness of 11-PATS.