Approximately locating an invisible agent in a graph with relative distance queries

TL;DR

This paper investigates how well a second player can locate an invisible moving agent in a graph using relative distance queries, providing bounds for specific graph classes and conjecturing a general logarithmic bound.

Contribution

It introduces a pursuit-evasion game with relative distance queries, establishing bounds for trees and grids, and conjecturing a logarithmic bound for all graphs.

Findings

Bounded the detection distance in trees of bounded degree and grids.

Proved that the detection distance is zero if the agent moves only at specific steps.

Conjectured a logarithmic bound for general graphs.

Abstract

In a pursuit evasion game on a finite, simple, undirected, and connected graph $G$, a first player visits vertices $m_1,m_2,\ldots$ of $G$, where $m_{i+1}$ is in the closed neighborhood of $m_i$ for every $i$, and a second player probes arbitrary vertices $c_1,c_2,\ldots$ of $G$, and learns whether or not the distance between $c_{i+1}$ and $m_{i+1}$ is at most the distance between $c_i$ and $m_i$. Up to what distance $d$ can the second player determine the position of the first? For trees of bounded maximum degree and grids, we show that $d$ is bounded by a constant. We conjecture that $d=O(\log n)$ for every graph $G$ of order $n$, and show that $d=0$ if $m_{i+1}$ may differ from $m_i$ only if $i$ is a multiple of some sufficiently large integer.