Given a finite set of integers $A$, a \emph{unary translocation} produces a new set $A' = A \cup \{u,v\}$, where $u$ and $v$ are nonnegative integers satisfying $x+y=u+v$ for some $x,y\in A$. For an input set $A$ and a target set $B$, the \emph{unary translocation distance} is the minimum number of unary translocations required to obtain a superset containing $B$. In this paper, we study this problem from both theoretical and computational perspectives. We prove that computing the unary translocation distance is strongly NP-hard, thereby answering an open question raised by \citet{ConstantinMiclausPopa2026UnaryTranslocation}. On the positive side, we give an exact pseudo-polynomial algorithm for every fixed constant value of $|B|$, extending our previous results for $|B|\leq 2$. For arbitrary target sets, we present a $2$-approximation algorithm, an additive $(|B|-1)$-approximation algorithm, and show that the additive algorithm also yields a $3$-approximation. We also propose parameterized algorithms, including algorithms parameterized by the maximum value in the input set together with the optimum distance, and by the maximum value in the target set together with $|B|$. In addition, we propose an integer linear programming formulation that gives an exact mathematical model for the problem, analyze its size, and show that the LP relaxation has integrality gap at least $\frac{4}{3}$. Finally, we report computational experiments comparing the $2$-approximation algorithm, beam search, and simulated annealing. The results show that the approximation algorithm is highly effective in practice and often outperforms the heuristic baselines.

翻译：给定一个有限整数集$A$，\emph{一元转位}操作生成新集合$A' = A \cup \{u,v\}$，其中$u$和$v$是非负整数，满足存在$x,y\in A$使得$x+y=u+v$。对于输入集合$A$和目标集合$B$，\emph{一元转位距离}定义为生成包含$B$的超集所需的最少一元转位次数。本文从理论与计算双重视角研究该问题。我们证明计算一元转位距离是强NP困难的，从而回答了\citet{ConstantinMiclausPopa2026UnaryTranslocation}提出的开放问题。在积极方面，针对$|B|$的任意固定常数值，我们给出精确的伪多项式算法，将之前针对$|B|\leq 2$的结果进行了推广。对于任意目标集合，我们提出一个$2$近似算法、一个加性$(|B|-1)$近似算法，并证明该加性算法同时实现$3$近似。我们还提出参数化算法，包括以输入集合最大值与最优距离为参数，以及以目标集合最大值与$|B|$为参数的算法。此外，我们提出整数线性规划形式化建模该问题，分析其规模，并证明LP松弛的整数间隙至少为$\frac{4}{3}$。最后，我们报告对比$2$近似算法、波束搜索和模拟退火的计算实验。结果表明，该近似算法在实际中高效且常优于启发式基线方法。