[Paper Review] Catching a mouse on a tree

Vytautas Gruslys, Arès Méroueh|arXiv (Cornell University)|Feb 23, 2015

Artificial Intelligence in Games3 references3 citations

TL;DR

This paper studies the minimum number of cats required to catch an invisible mouse on a tree graph, proving that ⌈(1/2)log₂n⌉ cats can always catch the mouse on any tree of order n, while also showing that (1/4−ε)log₂n cats are insufficient for some trees of order n. The key contribution is tightening the asymptotic bounds for the cat-and-mouse game on trees to Θ(log n).

ABSTRACT

In this paper we consider a pursuit-evasion game on a graph. A team of cats, which may choose any vertex of the graph at any turn, tries to catch an invisible mouse, which is constrained to moving along the vertices of the graph. Our main focus shall be on trees. We prove that $\lceil (1/2)\log_2(n) ceil$ cats can always catch a mouse on a tree of order $n$ and give a collection of trees where the mouse can avoid being caught by $ (1/4 - o(1))\log_2(n)$ cats.

Motivation & Objective

• To determine the minimum number of cats required to guarantee catching an invisible mouse on any tree of order n.
• To improve upon prior logarithmic upper and lower bounds for the cat-win number h(T) on trees.
• To establish tight asymptotic bounds by proving both an upper bound of ⌈(1/2)log₂n⌉ and a nearly-matching lower bound of (1/4−ε)log₂n cats.
• To analyze the structure of trees that maximize the number of cats required, using recursive decomposition and binary representations.

Proposed method

• Uses a recursive strategy based on the center of a tree, where removing a central vertex splits the tree into components of size at most ⌈(n−1)/2⌉.
• Applies induction to show that ⌈log₂n⌉ cats suffice by placing one cat at the center to block inter-component movement and using remaining cats to handle each subtree.
• Introduces a variant of the game on bipartite graphs where the mouse is restricted to one vertex class, enabling tracking via parity.
• Employs a binary tree structure B_k as a model to analyze worst-case trees, using vertex boundaries and edge subdivisions to simulate cat and mouse moves.
• Defines 'important vertices' and uses recursive component analysis to bound the number of safe positions the mouse can maintain.
• Applies induction on the size of the tree and uses bounds on the size of vertex boundaries and isolated components to show that the mouse can survive with fewer cats.

Experimental results

Research questions

• RQ1 What is the minimum number of cats required to guarantee catching a mouse on any tree of order n?
• RQ2 Can the upper bound of O(log n) cats be improved to a tight constant factor, such as (1/2)log₂n?
• RQ3 Is there a family of trees where the mouse can evade (1/4−ε)log₂n cats for arbitrarily large n?
• RQ4 How does the structure of the tree, particularly its recursive decomposition, affect the cat-win number h(T)?
• RQ5 What is the asymptotic growth rate of g(n) = max{h(T) : T is a tree of order n}?

Key findings

• The paper proves that ⌈(1/2)log₂n⌉ cats can always catch a mouse on any tree of order n, improving the prior upper bound of ⌈log₂n⌉.
• It constructs a family of trees where the mouse can evade (1/4−ε)log₂n cats for any ε>0 and sufficiently large n, showing that the lower bound is nearly tight.
• The growth rate of g(n), the maximum cat-win number over all trees of order n, is established as Θ(log n), resolving the asymptotic complexity of the problem.
• The paper improves upon the prior lower bound of c₂log n / log log n by Abramoskaya et al. to (1/4−ε)log n, significantly tightening the gap.
• The analysis uses recursive decomposition of trees and binary tree structures to simulate worst-case evasion strategies, with vertex boundaries and component isolation as key tools.
• The proof shows that with fewer than (1/4−ε)log₂n cats, the mouse can maintain at least n important vertices in its position set across even-numbered turns, ensuring indefinite survival.