Treasure Hunt in Graph Using Pebbles

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) and they are labeled arbitrarily as $0,1,\ldots , deg(v)-1$. At a node t in G a stationary object, called 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 time required to find the treasure is the total number of edges the agent visits before it finds the treasure. The agent neither have any prior knowledge about the graph nor the position of the treasure. An oracle that knows the graph, the agent’s initial position, 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 $\varDelta$ and diameter D if k pebbles are placed?”

We answer the above question when $k<D$ and $k=cD$ 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 $k<D$.

A. Bhattacharya—Supported by CSIR, Govt. of India, Grant Number: 09/731(0178)/2020-EMR-I

B. Gorain—Partially supported by SERB, Govt. of India (Grant Number: CRG/2020/005964 and Grant Number: MTR/2021/000118) and Research Initiation Grant supported by the IIT Bhilai, India.

P. S. Mandal—Partially supported by SERB, Govt. of India, Grant Number: MTR/2019/001528.