Treasure Hunt in Graph using Pebbles

TL;DR

This paper investigates the problem of a mobile agent finding a hidden treasure in a graph using pebbles as guides, analyzing how the number of pebbles affects the search time.

Contribution

It introduces algorithms and bounds for treasure hunt efficiency based on the number of pebbles, especially when pebbles are fewer than the graph's diameter.

Findings

Optimal algorithms for different pebble counts

Lower bounds for treasure hunt time with limited pebbles

Trade-offs between pebble number and search efficiency

Abstract

In this paper, we study the treasure hunt problem in a graph by a mobile agent. The nodes in the graph $G=(V,E)$ are anonymous and the edges incident to a vertex $v\in V$ whose degree is $deg(v)$ are labeled arbitrarily as $0,1,\ldots, deg(v)-1$. At a node $t$ in $G$ a stationary object, called {\it treasure} is located. The mobile agent that is initially located at a node $s$ in $G$, the starting point of the agent, must find the treasure by reaching the node $t$. The distance from $s$ to $t$ is $D$. The {\it time} required to find the treasure is the total number of edges the agent visits before it finds the treasure. The agent does not have any prior knowledge about the graph or the position of the treasure. An oracle, that knows the graph, the initial position of the agent, and the position of the treasure, places some pebbles on the nodes, at most one per node, of the graph to guide the agent towards the treasure. This paper aims to study the trade-off between the number of pebbles provided and the time required to find the treasure. To be specific, we aim to answer the following question. ``What is the minimum time for treasure hunt in a graph with maximum degree $\Delta$ and diameter $D$ if $k$ pebbles are placed? " We answer the above question when $k<D$ or $k=cD$ for some positive integer $c$. We design efficient algorithms for the agent for different values of $k$. We also propose an almost matching lower bound result for $k<D$.