k-Capture in Multiagent Pursuit Evasion, or the Lion and the Hyenas

TL;DR

This paper generalizes pursuit-evasion to k-capture, where at least k pursuers must simultaneously reach an evader in Euclidean space, establishing conditions for successful capture based on geometric configurations.

Contribution

It introduces the concept of k-capture in pursuit-evasion, providing necessary and sufficient conditions and constructive strategies for successful capture in Euclidean spaces.

Findings

k-capture is possible iff the evader lies in the interior of the pursuers' k-Hull.

In convex subsets, k-capture can always be achieved with a constructive strategy.

The study extends classical pursuit-evasion models to multi-pursuer scenarios.

Abstract

We consider the following generalization of the classical pursuit-evasion problem, which we call k-capture. A group of n pursuers (hyenas) wish to capture an evader (lion) who is free to move in an m-dimensional Euclidean space, the pursuers and the evader can move with the same maximum speed, and at least k pursuers must simultaneously reach the evader's location to capture it. If fewer than k pursuers reach the evader, then those pursuers get destroyed by the evader. Under what conditions can the evader be k-captured? We study this problem in the discrete time, continuous space model and prove that k-capture is possible if and only there exists a time when the evader lies in the interior of the pursuers' k-Hull. When the pursuit occurs inside a compact, convex subset of the Euclidean space, we show through an easy constructive strategy that k-capture is always possible.