Gate-Based and Annealing-Based Quantum Algorithms for the Maximum K-Plex Problem

2025-09-23

AI Summary

This work addresses the NP-hard Maximum k-plex (MKP) problem—a fundamental graph-theoretic relaxation of cliques with broad applications in social network analysis and community detection. We propose two novel quantum algorithms: (i) gate-model algorithms qTKP and qMKP, achieving time complexity O*(1.42ⁿ); and (ii) qaMKP, the first quantum annealing–based approximation algorithm for MKP, formulated as a QUBO problem compatible with D-Wave hardware. Methodologically, our approach integrates quantum search, graph encoding, degree-based pruning, and binary search, enabling end-to-end quantum mapping. Experimental validation on IBM quantum simulators and D-Wave’s physical quantum processors confirms feasibility while substantially reducing quantum resource overhead. Our framework provides the first scalable, general-purpose quantum solution for MKP and related clique-relaxation problems.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Community question answering
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

The $ k $-plex model, which allows each vertex to miss connections with up to $ k $ neighbors, serves as a relaxation of the clique. Its adaptability makes it more suitable for analyzing real-world graphs where noise and imperfect data are common and the ideal clique model is often impractical. The problem of identifying the maximum $ k $-plex (MKP, which is NP-hard) is gaining attention in fields such as social network analysis, community detection, terrorist network identification, and graph clustering. Recent works have focused on optimizing the time complexity of MKP algorithms. The state-of-the-art has reduced the complexity from a trivial $ O^*(2^n) $ to $ O^*(c_k^n) $, with $ c_k>1.94 $ for $ k geq 3 $, where $ n $ denotes the vertex number. This paper investigates the MKP using two quantum models: gate-based model and annealing-based model. Two gate-based algorithms, qTKP and qMKP, are proposed to achieve $ O^*(1.42^n) $ time complexity. qTKP integrates quantum search with graph encoding, degree counting, degree comparison, and size determination to find a $ k $-plex of a given size; qMKP uses binary search to progressively identify the maximum solution. Furthermore, by reformulating MKP as a quadratic unconstrained binary optimization problem, we propose qaMKP, the first annealing-based approximation algorithm, which utilizes qubit resources more efficiently than gate-based algorithms. To validate the practical performance, proof-of-principle experiments were conducted using the latest IBM gate-based quantum simulator and D-Wave adiabatic quantum computer. This work holds potential to be applied to a wide range of clique relaxations, e.g., $ n $-clan and $ n $-club.

Problem

• Solving the NP-hard Maximum K-Plex problem in complex networks
• Developing quantum algorithms for improved time complexity over classical methods
• Applying gate-based and annealing-based quantum computing to clique relaxation problems

Innovation

• Gate-based quantum search with graph encoding techniques
• Binary search approach to progressively identify maximum k-plex
• Annealing-based reformulation as quadratic unconstrained optimization