Hunting a rabbit: complexity, approximability and some characterizations

In the Hunters and Rabbit game, $k$ hunters attempt to shoot an invisible rabbit on a given graph $G$. In each round, the hunters select $k$ vertices to shoot at, while the rabbit moves 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 integer $k$ such that $k$ hunters have a winning strategy regardless of the rabbit's moves. The computational complexity of determining $h(G)$ has been one of the longest-standing open questions about the game. Our first main contribution resolves this by proving that computing $h(G)$ is NP-hard, even for bipartite simple graphs. We further show that the problem remains NP-hard even when $h(G) = O(n^{\epsilon})$ or when $n - h(G) = O(n^{\epsilon})$, where $n$ is the order of $G$. In addition, we prove that it is NP-hard to approximate $h(G)$ additively within $O(n^{1-\epsilon})$. When a time limit $l$ is imposed on the hunting process, we show that computing $h(G)$ remains NP-hard for any $l \ge 2$ bounded by a polynomial in $n$. On the positive side, we present a polynomial-time $l$-factor approximation algorithm for computing the hunting number with time limit $l$, and we show that $h(G)$ can be computed in polynomial time for bipartite graphs when only two time slots are allowed ($l = 2$). Finally, we provide a forbidden-subgraph characterization of graphs with loops that satisfy $h(G) = 1$, extending a known characterization for simple graphs.