二值模式瓦片集合成是NP难的
Binary pattern tile set synthesis is NP-hard
摘要：
在算法自组装领域，一个长期存在的未被证明的猜想是二进制模式瓦片集合成（2-PATS）的NP-困难。$k$ -PATS问题是设计具有最小数量的拼块类型的拼块组装系统，该拼块组装系统将自组装$k$颜色的输入图案。具有理论和实际意义的$k$已经在一系列论文中进行了研究，这些论文已经表明$k$对于$k = 60$、$k = 29$、然后是$k = 11$是NP难的。在本文中，我们关闭的基本猜想，2-PATS是NP-困难的，总结这条研究路线。虽然我们的大多数证明依赖于标准的数学证明技术，但一个关键的引理使用了计算机辅助证明，这是一种相对新颖但越来越多地用于推导复杂数学问题证明的范式。这一工具在解决组合问题时特别强大，例如阿佩尔和哈肯对四色定理的证明（后来由罗伯逊、桑德斯、西摩和托马斯简化），以及科涅夫和利西察使用计算机程序在埃尔德什差异问题上的最新重要进展。我们利用一个大规模的并行算法，从而把一个棘手的部分，我们的证明到一个程序，需要大约一年的计算时间，使使用计算机辅助证明到一个新的规模。我们充分详细说明了我们的代码所采用的算法，并使代码免费在线提供。
英文摘要：
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.
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