Hunting a rabbit: complexity, approximability and some characterizations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908692119552000 |
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| author | Ben-Ameur, Walid Gahlawat, Harmender Maddaloni, Alessandro |
| author_facet | Ben-Ameur, Walid Gahlawat, Harmender Maddaloni, Alessandro |
| contents | 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 $h(G) = O(n^ε)$ or when $n - h(G) = O(n^ε)$, where $n$ is the order of $G$. In addition, we prove that it is NP-hard to approximate $h(G)$ additively within $O(n^{1-ε})$.
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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_15982 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hunting a rabbit: complexity, approximability and some characterizations Ben-Ameur, Walid Gahlawat, Harmender Maddaloni, Alessandro Combinatorics Computational Complexity 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 $h(G) = O(n^ε)$ or when $n - h(G) = O(n^ε)$, where $n$ is the order of $G$. In addition, we prove that it is NP-hard to approximate $h(G)$ additively within $O(n^{1-ε})$. 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. |
| title | Hunting a rabbit: complexity, approximability and some characterizations |
| topic | Combinatorics Computational Complexity |
| url | https://arxiv.org/abs/2502.15982 |