Hunting a rabbit is hard

Abstract: In the Hunters and Rabbit game, $k$ hunters attempt to shoot an invisible rabbit on a given graph $G$. In each round, the hunters can choose $k$ vertices to shoot at, while the rabbit must move along an edge of $G$. The hunters win if at any point the rabbit is shot. The hunting number of $G$, denoted $h(G)$, is the minimum $k$ for which $k$ hunters can win, regardless of the rabbit's moves. The complexity of computing $h(G)$ has been the longest standing open problem concerning the game and has been posed as an explicit open problem by several authors. The first contribution of this paper resolves this question by establishing that computing $h(G)$ is NP-hard even for bipartite simple graphs. We also prove that the problem remains hard even when $h(G)$ is $O(n^{\epsilon})$ or when $n-h(G)$ is $O(n^{\epsilon})$, where $n$ is the order of $G$. Furthermore, we prove that it is NP-hard to additively approximate $h(G)$ within $O(n^{1-\epsilon})$. Finally, we give a characterization of graphs with loops for which $h(G)=1$ by means of forbidden subgraphs, extending a known characterization for simple graphs.