The uniqueness of an optimal solution to a combinatorial optimization problem attracts many fields of researchers' attention because it has a wide range of applications, it is related to important classes in computational complexity, and an instance with only one solution is often critical for algorithm designs in theory. However, as the authors know, there is no major benchmark set consisting of only instances with unique solutions, and no algorithm generating instances with unique solutions is known; a systematic approach to getting a problem instance guaranteed having a unique solution would be helpful. A possible approach is as follows: Given a problem instance, we specify a small part of a solution in advance so that only one optimal solution meets the specification. This paper formulates such a ``pre-assignment'' approach for the vertex cover problem as a typical combinatorial optimization problem and discusses its computational complexity. First, we show that the problem is $\Sigma^P_2$-complete in general, while the problem becomes NP-complete when an input graph is bipartite. We then present an $O(2.1996^n)$-time algorithm for general graphs and an $O(1.9181^n)$-time algorithm for bipartite graphs, where $n$ is the number of vertices. The latter is based on an FPT algorithm with $O^*(3.6791^{\tau})$ time for vertex cover number $\tau$. Furthermore, we show that the problem for trees can be solved in $O(1.4143^n)$ time.

翻译：组合优化问题的唯一最优解因其广泛的应用、与计算复杂性中重要类的关联，以及仅含单一解的实例对算法理论设计的关键性，吸引了众多领域研究者的关注。然而，据作者所知，目前尚不存在仅由唯一解实例构成的主要基准测试集，也缺乏生成唯一解实例的已知算法；因此，建立一种系统化方法以确保获得具有唯一解的实例将具有重要价值。一种可行方案是：给定问题实例后，预先指定解的一小部分，使得仅有一个最优解满足该规范。本文针对典型组合优化问题中的顶点覆盖问题，形式化了这种"预分配"方法，并探讨其计算复杂度。首先，我们证明该问题在一般情况下为$\Sigma^P_2$完全问题，而当输入图为二部图时则变为NP完全问题。随后，我们提出针对一般图的$O(2.1996^n)$时间算法以及针对二部图的$O(1.9181^n)$时间算法（其中$n$为顶点数）。后者基于顶点覆盖数$\tau$的$O^*(3.6791^{\tau})$时间FPT算法。此外，我们证明树图上的该问题可在$O(1.4143^n)$时间内求解。