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IEEE International Conference on Data Engineering (ICDE)

Quantum Algorithms for the Maximum K-Plex Problem

Xiaofan Li, Gao Cong, Rui Hong Zhou

Published May 13, 2024• 2 citations
Publisher

Abstract

The k-plex model, which allows each vertex to miss connections with up to$k$neighbors, serves as a relaxation of the clique model. Its adaptability makes it more suitable for analyzing graphs from real-world applications, where noise and imperfect data are common and the stringent clique model is often impractical. The challenge of identifying maximum k-plex (MKP, an NP-hard problem) is gaining attention in fields such as social network analysis, community detection, terrorist network identification, and graph clustering. Recent research efforts 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$k$> 3, where$n$denotes the number of vertices. In this paper, we demonstrate that MKP can be solved in$O^{*}(1.42^{n})$and propose the first two quantum algorithms, qTKP and qMKP, to achieve this complexity. qTKP employs quantum search integrated with graph encoding, degree count, degree comparison, and size determination to find a k-plex of a given size; qMKP uses a binary search to progressively identify the maximum solution. To validate the practical performance and effectiveness of our algorithms, proof-of-principle experiments were conducted using the latest IBM quantum simulator currently available. This work holds potential to be applied to a wide range of clique relaxations, e.g., n-clan and n-club.

DOI
10.1109/icde60146.2024.00192

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The k-plex model, which allows each vertex to miss connections with up tokkkneighbors, serves as a relaxation of the clique model. Its adaptability makes it more suitable for analyzing graphs from real-world applications, where noise and imperfect data are common and the stringent clique model is often impractical. The challenge of identifying maximum k-plex (MKP, an NP-hard problem) is gaining attention in fields such as social network analysis, community detection, terrorist network identification, and graph clustering. Recent research efforts have focused on optimizing the time complexity of MKP algorithms. The state-of-the-art has reduced the complexity from a trivialO∗(2n)O^{*}(2^{n})O∗(2n)toO∗(ckn)O^{*}(c_{k}^{n})O∗(ckn​), withck>1.94c_{k} > 1.94ck​>1.94forkkk> 3, wherennndenotes the number of vertices. In this paper, we demonstrate that MKP can be solved inO∗(1.42n)O^{*}(1.42^{n})O∗(1.42n)and propose the first two quantum algorithms, qTKP and qMKP, to achieve this complexity. qTKP employs quantum search integrated with graph encoding, degree count, degree comparison, and size determination to find a k-plex of a given size; qMKP uses a binary search to progressively identify the maximum solution. To validate the practical performance and effectiveness of our algorithms, proof-of-principle experiments were conducted using the latest IBM quantum simulator currently available. This work holds potential to be applied to a wide range of clique relaxations, e.g., n-clan and n-club.
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.
Identifying a biclique with the maximum number of edges bears considerable implications for numerous fields of application, such as detecting anomalies in E-commerce transactions, discerning protein-protein interactions in biology, and refining the efficacy of social network recommendation algorithms. However, the inherent NP-hardness of this problem significantly complicates the matter. The prohibitive time complexity of existing algorithms is the primary bottleneck constraining the application scenarios. Aiming to address this challenge, we present an unprecedented exploration of a quantum computing approach. Efficient quantum algorithms, as a crucial future direction for handling NP-hard problems, are presently under intensive investigation, of which the potential has already been proven in practical arenas such as cybersecurity. However, in the field of quantum algorithms for graph databases, little work has been done due to the challenges presented by the quantum representation of complex graph topologies. In this study, we delve into the intricacies of encoding a bipartite graph on a quantum computer. Given a bipartite graph with n vertices, we propose a ground-breaking algorithm qMBS with time complexity O^*(2^(n/2)), illustrating a quadratic speed-up in terms of complexity compared to the state-of-the-art. Furthermore, we detail two variants tailored for the maximum vertex biclique problem and the maximum balanced biclique problem. To corroborate the practical performance and efficacy of our proposed algorithms, we have conducted proof-of-principle experiments utilizing IBM quantum simulators, of which the results provide a substantial validation of our approach to the extent possible to date.
The maximal-clique problem, to find the maximally sized clique in a given graph, is classically an NP-complete computational problem, which has potential applications ranging from electrical engineering, computational chemistry, and bioinformatics to social networks. Here we develop a quantum algorithm to solve the maximal-clique problem for any graph GGG with nnn vertices with quadratic speedup over its classical counterparts, where the time and spatial complexities are reduced to, respectively, O(2n)O(\sqrt{{2}^{n}})O(2n​) and O(n2)O({n}^{2})O(n2). With respect to oracle-related quantum algorithms for the NP-complete problems, we identify our algorithm as optimal. To justify the feasibility of the proposed quantum algorithm, we successfully solve a typical clique problem for a graph GGG with two vertices and one edge by carrying out a nuclear magnetic resonance experiment involving four qubits.
Abstract The maximum clique problem (MCP) is to find the largest complete subgraph in an undirected graph, that is, the subgraph in which there are edges between every two different vertices. It is an NP-Hard problem with wide applications, including bioinformatics, social networks, data mining, and other fields. This paper proposes an enhanced quantum solution method that introduces a pre-detection phase based on a hybrid heuristic algorithm and a vertex pruning strategy, encoding the prior constraints on clique size into quantum registers as global information. It encodes the prior constraints on clique size into quantum registers as global information. In contrast, state-of-the-art Grover-based methods for n-vertex graphs require O ( n 2 n ) iterations and O ( n ) measurements. By leveraging a global information encoding mechanism, the proposed method compresses the number of iterations to O ( 2 n ) and only needs a constant number of measurements to identify all maximum cliques. We validate algorithmic correctness through simulations on IBM’s Qiskit platform and benchmark qubit/gate efficiency against existing Grover-based MCP solvers.
Abstract The maximum clique problem(MCP) is to find the largest complete subgraph in an undirected graph, that is, the subgraph in which there are edges between every two different vertices. It is an NP-Hard problem with wide applications, including bioinformatics, social networks, data mining, and other fields. This paper proposes an improved algorithm that dynamically tracks the maximum clique size by encoding prior constraints on the vertex count—derived from Turán's theorem and complete graph properties—into global variables through quantum circuit pre-detection. The algorithm further synergizes with Grover's search to optimize the solution space. Our auxiliary-qubit encoding scheme dynamically tracks clique sizes during quantum search, eliminating iterative measurements, achieving MCP solution with $O\left ( \sqrt{2^{n } } \right ) $ Grover iterations and $O\left ( 1 \right ) $ measurements. This represents an n\boldsymbol{n}n-fold improvement over state-of-the-art Grover-based methods, which require O(n2n)O(n\sqrt{2^n})O(n2n​) iterations and O(n)O(n)O(n) measurements for nnn-vertex graphs. We validate algorithmic correctness through simulations on IBM's Qiskit platform and benchmark qubit/gate efficiency against existing Grover-based MCP solvers.
This paper addresses the problem of finding the densest kkk-vertex subgraph in an arbitrary graph. This problem is NP-hard and has important applications in social network analysis, fraud detection, recommendation systems, and bioinformatics. We propose two quantum approaches to solve this problem: reduction to Quadratic Unconstrained Binary Optimization (QUBO) and using Grover's quantum search algorithm. For the latter approach, we present an explicit gate-based oracle circuit utilizing Dicke states and Quantum Fourier Transform for edge counting. Numerical simulations demonstrate a quadratic speedup over classical Brute-force search.