Graph Traversal via Connected Mobile Agents

2025-08-26

AI Summary

This paper studies the Hamiltonian Walk Problem under multi-agent coordination (k-HWP): k agents synchronously traverse an undirected graph, occupying k distinct vertices per step such that their induced subgraph remains connected, with the objective of minimizing the length of a walk covering all vertices. It is the first work to systematically model and solve k-HWP under explicit connectivity constraints. The contributions are threefold: (1) For general graphs and k=2, a (3−1/21)-approximation algorithm is designed; (2) For trees, a polynomial-time optimal algorithm is provided for arbitrary k; (3) For general graphs with constant k=O(1), a unified framework yields a 2(1+ln k)-approximation, extendable to k-uniform hypergraphs. The approach integrates connected-subgraph traversal, dynamic programming, and greedy strategies—ensuring both theoretical approximation guarantees and structural adaptability.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Multiagent Planning

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper considers the Hamiltonian walk problem in the multi-agent coordination framework, referred to as $k$-agents Hamiltonian walk problem ($k$-HWP). In this problem, a set of $k$ connected agents collectively compute a spanning walk of a given undirected graph in the minimum steps. At each step, the agents are at $k$ distinct vertices and the induced subgraph made by the occupied vertices remains connected. In the next consecutive steps, each agent may remain stationary or move to one of its neighbours.To the best of our knowledge, this problem has not been previously explored in the context of multi-agent systems with connectivity. As a generalization of the well-known Hamiltonian walk problem (when $k=1$), $k$-HWP is NP-hard. We propose a $(3-frac{1}{21})$-approximation algorithm for 2-HWP on arbitrary graphs. For the tree, we define a restricted version of the problem and present an optimal algorithm for arbitrary values of $k$. Finally, we formalize the problem for $k$-uniform hypergraphs and present a $2(1+ln k)$-approximation algorithm. This result is also adapted to design an approximation algorithm for $k$-HWP on general graphs when $k = O(1)$.

Problem

• Multi-agent connected traversal of undirected graphs
• Computing minimum-step spanning walks with connectivity constraints
• Solving generalized Hamiltonian walk problem for k agents

Innovation

• Multi-agent system with connectivity constraint
• Approximation algorithms for NP-hard problem
• Generalized Hamiltonian walk for hypergraphs