One of the most fundamental and well-studied problems in Tile Self-Assembly is the Unique Assembly Verification (UAV) problem. This algorithmic problem asks whether a given tile system uniquely assembles a specific assembly. The complexity of this problem in the 2-Handed Assembly Model (2HAM) at a constant temperature is a long-standing open problem since the model was introduced. Previously, only membership in the class coNP was known and that the problem is in P if the temperature is one ($\tau=1$). The problem is known to be hard for many generalizations of the model, such as allowing one step into the third dimension or allowing the temperature of the system to be a variable, but the most fundamental version has remained open. In this paper, we prove the UAV problem in the 2HAM is hard even with a small constant temperature ($\tau = 2$), and finally answer the complexity of this problem (open since 2013). Further, this result proves that UAV in the staged self-assembly model is coNP-complete with a single bin and stage (open since 2007), and that UAV in the q-tile model is also coNP-complete (open since 2004). We reduce from Monotone Planar 3-SAT with Neighboring Variable Pairs, a special case of 3SAT recently proven to be NP-hard. We accompany this reduction with a positive result showing that UAV is solvable in polynomial time with the promise that the given target assembly will have a tree-shaped bond graph, i.e., contains no cycles. We provide a $\mathcal{O}(n^5)$ algorithm for UAV on tree-bonded assemblies when the temperature is fixed to $2$, and a $\mathcal{O}(n^5\log \tau)$ time algorithm when the temperature is part of the input.
翻译:Tile Self- Assembly 中最根本和研究周密的问题之一是 UAV 的 UAV 问题。 这个算法问题询问给定的牌子系统是否具有独特的组合。 2HAM 模式在恒定温度下存在这一问题的复杂性是自该模式推出以来长期存在的问题。 之前, 只知道该类公司的成员身份, 而如果温度为1美元( tau=1美元), 问题在于P 。 这个模型的很多常规化问题众所周知, 例如允许一个步骤进入第三个维度, 或允许这个系统的承诺周期是一个可变数, 但最基本的版本仍然开放。 在本文中, 我们证明 2HAM 2HA 的UAV 问题即使有一个小的恒定温度( tau=2美元) 也很难解决, 最后解答这个问题的复杂性( 自2013年起开始开放 ) 。 这个结果证明, 级的UAVO- sal- sal- sal- silver 模式是单一的硬值和阶值, 自2007年起开始, AS- pal- preal- preal- pal- pal- pal- pasion a case a case a pro- pro- pro- pro- ex- pro- pro- putmental) ex- deal- exional- put laveal- exional- ex- ex- ex- exion a realutmentalutmentalutmental- lautment a a a real- palut dealut the a pal- ex- putmentald.