Vertex Cover Interdiction in Bipartite Graphs

Abstract

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, Gr\"une and Wulf proved that the problem is $\Sigma_2^p$-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 $k$.