Hunting a rabbit is hard

2025-02-21

AI Summary

This paper addresses the long-standing open problem of determining the computational complexity of the hunting number $h(G)$ of a graph $G$—the minimum number of hunters required to guarantee capturing an invisible rabbit in the hunter–rabbit game. Via a carefully constructed NP-hardness reduction, we establish, for the first time, that computing $h(G)$ remains NP-hard even on bipartite graphs. Furthermore, we prove strong inapproximability: unless P = NP, no additive $O(n^{1-varepsilon})$-approximation algorithm exists for any $varepsilon > 0$. Additionally, we provide a complete forbidden-subgraph characterization for graphs with $h(G) = 1$: $h(G) = 1$ if and only if $G$ excludes all six explicitly specified induced subgraphs. Collectively, these results resolve the three central questions concerning the hunting number—its computational complexity, approximability, and structural characterization—unifying prior fragmented understanding.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

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.

Problem

• Computing hunting number is NP-hard
• Hardness holds for bipartite simple graphs
• Characterization of graphs with hunting number one

Innovation

• NP-hard computation
• additive approximation
• forbidden subgraphs characterization