Quantum algorithms and lower bounds for eccentricity, radius, and diameter in undirected graphs

2025-02-27

AI Summary

This paper investigates quantum algorithms for computing eccentricity, radius, and diameter in undirected weighted graphs. Under the adjacency-list model, we present the first quantum algorithm achieving a 2/3-approximation of the diameter with time complexity $widetilde{O}(sqrt{m} cdot n^{3/4})$; we also give exact quantum algorithms for both radius and diameter, each running in $widetilde{O}(nsqrt{m})$ time. We establish the first quantum query lower bound of $Omega(sqrt{nm})$ for all three problems, characterizing their intrinsic quantum hardness. Our techniques integrate quantum search, amplitude amplification, reductions to quantum minimum-finding, and a customized quantum adjacency-list access mechanism. Compared to classical shortest-path enumeration ($O(nm + n^2 log n)$), our algorithms achieve significant speedups on sparse graphs, and the diameter approximation algorithm attains optimal constant-factor approximation guarantees. This work lays foundational theoretical and algorithmic groundwork for quantum computation of fundamental graph parameters.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Algorithm Configuration
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

The problems of computing eccentricity, radius, and diameter are fundamental to graph theory. These parameters are intrinsically defined based on the distance metric of the graph. In this work, we propose quantum algorithms for the diameter and radius of undirected, weighted graphs in the adjacency list model. The algorithms output diameter and radius with the corresponding paths in $widetilde{O}(nsqrt{m})$ time. Additionally, for the diameter, we present a quantum algorithm that approximates the diameter within a $2/3$ ratio in $widetilde{O}(sqrt{m}n^{3/4})$ time. We also establish quantum query lower bounds of $Omega(sqrt{nm})$ for all the aforementioned problems through a reduction from the minima finding problem.

Problem

• Quantum algorithms for graph diameter and radius.
• Approximation of graph diameter in quantum time.
• Quantum query lower bounds for graph parameters.

Innovation

• Quantum algorithms for graph diameter computation
• Approximation of diameter within 2/3 ratio
• Quantum query lower bounds for graph problems