发表机构
Graduate School of Information Sciences, Tohoku University; Faculty of Science and Technology, Keio University; Graduate School of Informatics, Kyoto University; Research Institute for Mathematical Sciences, Kyoto University; Graduate School of Informatics and Engineering, The University of Electro-Communications（东北大学大学院信息科学研究科; 庆应义塾大学理工学部; 京都大学情报学府; 京都大学数理物理学研究所; 电气通信大学情报工学研究科）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究二分图中的顶点覆盖阻断问题，证明其NP-完全性，但限制于最小顶点覆盖时可多项式求解，并分析参数化复杂性，给出W[1]-难和固定参数可处理的结果。
AI 中文摘要
在顶点覆盖阻断问题中，给定一个无向图$G=(V,E)$、两个整数$t$和$k$以及一个顶点子集$B\subseteq V$，要求找到一个集合$X \subseteq B$，满足$|X|\leq t$，使得$X$击中（即相交）$G$中所有大小至多为$k$的顶点覆盖。最近，Grüne和Wulf证明了该问题是$\Sigma_2^p$-完全的。然而，他们的归约依赖于顶点覆盖问题是NP-完全的。这反过来意味着，当输入图限制为二分图时，我们不知道顶点覆盖阻断问题的复杂性状态，因为对于二分图，顶点覆盖问题可以在多项式时间内解决。我们的主要结果之一表明，对于二分图，顶点覆盖阻断问题是NP-完全的。相反，当$k$被限制为最小顶点覆盖大小，即我们仅要求击中所有最小顶点覆盖时，我们表明对于二分图，顶点覆盖阻断问题可以在多项式时间内解决。这促使我们研究当$k$与最小顶点覆盖大小的差值作为参数时，二分图上顶点覆盖阻断问题的参数化复杂性。使用该参数，我们表明该问题是$\mathrm{W}[1]$-难的，但当参数为常数时（即XP时间），可以在多项式时间内解决。我们还表明，当以$k$为参数时，该问题是固定参数可处理的。
英文摘要
In the vertex cover interdiction problem, we are given an undirected graph $G=(V,E)$, two integers $t$ and $k$ and a vertex subset $B\subseteq V$, and we are asked to find a set $X \subseteq B$ with $|X|\leq t$ such that $X$ hits (i.e., intersects) all the vertex covers of $G$ of size at most $k$. Recently, Grüne and Wulf proved that the problem is $Σ_2^p$-complete. However, their reduction relied on the fact that the vertex cover problem is NP-complete. This, in turn, means that we do not know the complexity status of the vertex cover interdiction problem when the input graph is restricted to a bipartite graph since the vertex cover problem can be solved in polynomial time for bipartite graphs. One of our main results shows that the vertex cover interdiction problem is NP-complete for bipartite graphs. In contrast, when $k$ is restricted to the minimum vertex cover size, i.e., we are only required to hit all the minimum vertex covers, we show that the vertex cover interdiction problem can be solved in polynomial time for bipartite graphs. This motivates us to study the parameterized complexity of the vertex cover interdiction problem for bipartite graphs when the difference of $k$ and the minimum vertex cover size is taken as a parameter. With this parameter, we show that the problem is $\mathrm{W}[1]$-hard, but can be solved in polynomial time when the parameter is constant (i.e., in XP time). We also show that the problem is fixed-parameter tractable when parameterized by $k$.
CommentsISAAC 2026
