Binary Pattern Tile Set Synthesis Is NP-Hard

Abstract

In the field of algorithmic self-assembly, a long-standing unproven conjecture has been that of the NP-hardness of binary pattern tile set synthesis (2-Pats). The k-Pats problem is that of designing a tile assembly system with the smallest number of tile types which will self-assemble an input pattern of k colors. Of both theoretical and practical significance, k-Pats has been studied in a series of papers which have shown k-Pats to be NP-hard for k = 60, k = 29, and then k = 11. In this paper, we close the fundamental conjecture that 2-Pats is NP-hard, concluding this line of study.While most of our proof relies on standard mathematical proof techniques, one crucial lemma makes use of a computer-assisted proof, which is a relatively novel but increasingly utilized paradigm for deriving proofs for complex mathematical problems. This tool is especially powerful for attacking combinatorial problems, as exemplified by the proof of the four color theorem by Appel and Haken (simplified later by Robertson, Sanders, Seymour, and Thomas) or the recent important advance on the Erdős discrepancy problem by Konev and Lisitsa using computer programs. We utilize a massively parallel algorithm and thus turn an otherwise intractable portion of our proof into a program which requires approximately a year of computation time, bringing the use of computer-assisted proofs to a new scale. We fully detail the algorithm employed by our code, and make the code freely available online.

Crazy Sequential Representations of Numbers for Small Bases

“…A brute force approach to a problem like this has generally been classified as computational mathematics -there is a point for many problems at which the number of possible combinations becomes too large for a human, or humanity, to check by hand in any reasonable amount of time. This has become more common with efforts to verify and prove other long open questions in mathematics such as the Kepler Conjecture [5,6,7], the Boolean Pythagorean Triples problem [8], finding Ramsey numbers [3,14,15], the Happy Ending problem [11,17], the 2-PATS problem [9], and many others where brute-force exhaustive-search solutions were required.…”

“…This recurrence relation yields an efficient solution allowing an exhaustive examination within a reasonable amount of time. For most of the bases in our study (3)(4)(5)(6)(7)(8)(9), even basic laptops are sufficient to check the millions of combinations. For base ten, we utilized some research servers due to the high memory requirements.…”

“…for all y ∈ RS do 13: (10,1,9) tion of terms, evaluation in the given base, and processing such large amounts of data. We cover these in the analysis of Algorithm 1, which is a dynamic programming solution to the problem.…”

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

“…Johnsen, Kao, and Seki strengthened the result up to the NP-hardness of 29-Pats with 47/46 ≈ 1.022 being an approximation ratio unachievable in polynomial time, unless P = NP [4]. Kari et al have recently proposed a computer-assisted proof for the NPhardness of 2-Pats [5]. As a corollary of their proof, the approximation ratio 14/13 ≈ 1.077 is proven polynomial-time unachievable.…”

“…In contrast, (T, F, T, F) does not satisfy φ in the 1-in-3-Sat sense because it satisfies more than one literal of the first clause 5. n does not mean a negated variable.…”