Given a graph G on n vertices and two integers k and d, the Contraction(vc) problem asks whether one can contract at most k edges to reduce the vertex cover number of G by at least d. Recently, Lima et al. [JCSS 2021] proved that Contraction(vc) admits an XP algorithm running in time $f(d)\cdot n^{\mathcal{O}(d)}$. They asked whether this problem is FPT under this parameterization. In this article, we prove that: (i) Contraction(vc) is W[1]-hard parameterized by $k+d$. Moreover, unless the ETH fails, the problem does not admit an algorithm running in time $f(k+d)\cdot n^{o(k+d)}$ for any function f. This answers negatively the open question stated in Lima et al. [JCSS 2021]. (ii) Contraction(vc) is NP-hard even when $k=d$. (iii) Contraction(vc) can be solved in time $2^{\mathcal{O}(d)}\cdot n^{k-d+\mathcal{O}(1)}$. This improves the algorithm of Lima et al. [JCSS 2021], and shows that when $k=d$, Contraction(vc) is FPT parameterized by d (or by k).

给定一个包含 n 个顶点和两个整数 k 和 d 的图G ， Contraction ( vc )问题询问是否可以收缩至多k条边以将G的顶点覆盖数至少减少d。最近，利马等人。[JCSS 2021] 证明Contraction( vc )承认一个XP算法及时运行$F(d)\cdot n^{\mathcal{欧}(d)}$. 他们问这个问题是不是这个参数化下的FPT。在本文中，我们证明：(i) Contraction( vc )是W [1]-硬参数化$k+d$. 此外，除非ETH失败，否则该问题不会承认及时运行的算法$F(k+d)\cdot n^{o(k+d)}$对于任何函数f。这否定了利马等人提出的开放性问题。[JCSS 2021]。(ii)收缩 ( vc )是NP -即使当$k=d$. (iii) Contraction( vc )可以及时解决${\mathrm{2个}}^{\mathcal{欧}(d)}\cdot n^{k-d+\mathcal{欧}(\mathrm{1个})}$. 这改进了 Lima 等人的算法。[JCSS 2021]，并表明当$k=d$, Contraction( vc )是由d（或k ）参数化的FPT。