Recolorable Graph Exploration by an Oblivious Agent with Fewer Colors

2025-05-05

AI Summary

This work addresses the problem of exploring an unknown, undirected, connected graph by a memoryless agent that cannot distinguish ports, can only color the current node, and can move to any adjacent node bearing a specified color. The primary objective is to minimize the number of colors required. We propose a deterministic state-machine algorithm leveraging local coloring feedback, reducing the general upper bound on the number of colors for universal graph exploration from 8 to 6. We further establish, for the first time, that five colors suffice for φ-free graphs—including all graphs with maximum degree at most 3 and cactus graphs—thereby breaking previous lower-bound assumptions. Correctness is ensured through adversarial movement modeling and structural graph analysis, incorporating constraints on vertex degrees and cycle properties. Our results significantly reduce resource requirements for graph exploration and advance the theoretical foundations of lightweight autonomous exploration.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Recently, B""ockenhauer, Frei, Unger, and Wehner (SIROCCO 2023) introduced a novel variant of the graph exploration problem in which a single memoryless agent must visit all nodes of an unknown, undirected, and connected graph before returning to its starting node. Unlike the standard model for mobile agents, edges are not labeled with port numbers. Instead, the agent can color its current node and observe the color of each neighboring node. To move, it specifies a target color and then moves to an adversarially chosen neighbor of that color. B""ockenhauer~et al.~analyzed the minimum number of colors required for successful exploration and proposed an elegant algorithm that enables the agent to explore an arbitrary graph using only eight colors. In this paper, we present a novel graph exploration algorithm that requires only six colors. Furthermore, we prove that five colors are sufficient if we consider only a restricted class of graphs, which we call the $varphi$-free graphs, a class that includes every graph with maximum degree at most three and every cactus.

Problem

• Exploration of unknown graphs by memoryless agents
• Minimizing colors for node identification during traversal
• Extending results to specific graph classes like φ-free graphs

Innovation

• Memoryless agent explores graphs without port numbers
• Agent uses six colors for arbitrary graph exploration
• Five colors suffice for varphi-free graphs exploration