Minimum sum vertex cover of an $n$-vertex graph $G$ is a bijection $\phi : V(G) \to [n]$ that minimizes the cost $\sum_{\{u,v\} \in E(G)} \min \{\phi(u), \phi(v) \}$. Finding a minimum sum vertex cover of a graph (the MSVC problem) is NP-hard. MSVC is studied well in the realm of approximation algorithms. The best-known approximation factor in polynomial time for the problem is $16/9$ [Bansal, Batra, Farhadi, and Tetali, SODA 2021]. Recently, Stankovic [APPROX/RANDOM 2022] proved that achieving an approximation ratio better than $1.014$ for MSVC is NP-hard, assuming the Unique Games Conjecture. We study the MSVC problem from the perspective of parameterized algorithms. The parameters we consider are the size of a minimum vertex cover and the size of a minimum clique modulator of the input graph. We obtain the following results. 1. MSVC can be solved in $2^{2^{O(k)}} n^{O(1)}$ time, where $k$ is the size of a minimum vertex cover. 2. MSVC can be solved in $f(k)\cdot n^{O(1)}$ time for some computable function $f$, where $k$ is the size of a minimum clique modulator.

翻译：给定一个包含$n$个顶点的图$G$，最小和顶点覆盖是指双射$\phi : V(G) \to [n]$，使得代价$\sum_{\{u,v\} \in E(G)} \min \{\phi(u), \phi(v) \}$最小化。求解图的最小和顶点覆盖（MSVC问题）是NP难的。MSVC在近似算法领域已得到充分研究。该问题在多项式时间内已知最佳近似因子为$16/9$ [Bansal, Batra, Farhadi, and Tetali, SODA 2021]。近期，Stankovic [APPROX/RANDOM 2022]证明，假设唯一游戏猜想成立，MSVC的近似比优于$1.014$是NP难的。我们从参数化算法的角度研究MSVC问题。考虑的参数包括输入图的最小顶点覆盖大小和最小团调制器大小。我们获得以下结果：1. MSVC可在$2^{2^{O(k)}} n^{O(1)}$时间内求解，其中$k$为最小顶点覆盖大小。2. MSVC可在$f(k)\cdot n^{O(1)}$时间内求解，其中$f$为某个可计算函数，$k$为最小团调制器大小。