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Scalable Quantum Walk-Based Heuristics for the Minimum Vertex Cover Problem

F. S. Luiz, A. K. F. Iwakami, D. H. Moraes, M. C. de Oliveira

TL;DR

This work addresses the NP-hard Minimum Vertex Cover problem by introducing a scalable, resource-efficient heuristic based on continuous-time quantum walks. By encoding vertices with $\lceil \log_2(V)\rceil$ qubits and evolving under a graph-encoded normalized Laplacian Hamiltonian, the method uses short-time CTQW dynamics to identify influential vertices via transition probabilities, followed by a dynamic decoupling (freezing) step to remove selected vertices from further interference. The approach achieves superior approximation ratios across diverse graph ensembles (ER, BA, REG) and exhibits robustness to topology, outperforming classical heuristics and MILP baselines in many cases, while benefiting from an exponential qubit reduction. This topology-agnostic, quantum-inspired framework has potential implications for large-scale network optimization and related applications in resilience, containment, and infrastructure planning, with feasible pathways toward quantum or hybrid implementations.

Abstract

We propose a novel heuristic quantum algorithm for the Minimum Vertex Cover (MVC) problem based on continuous-time quantum walks (CTQWs). In this framework, the coherent propagation of a quantum walker over a graph encodes its structural properties into state amplitudes, enabling the identification of highly influential vertices through their transition probabilities. To enhance stability and solution quality, we introduce a dynamic decoupling (``freezing'') mechanism that isolates vertices already selected for the cover, preventing their interference in subsequent iterations of the algorithm. The method employs a compact binary encoding, requiring only $\lceil \log_2 (V)\rceil$ qubits to represent a graph with $V$ vertices, resulting in an exponential reduction of quantum resources compared to conventional vertex-based encodings. We benchmark the proposed heuristic against exact solutions obtained via Mixed-Integer Linear Programming (MILP) and against established classical heuristics, including Simulated Annealing, FastVC, and the 2-Approximation algorithm, across Erdős--Rényi, Barabási--Albert and regular random graph ensembles. Our results demonstrate that the CTQW-based heuristic consistently achieves superior approximation ratios and exhibits remarkable robustness with respect to network topology, outperforming classical approaches in both heterogeneous and homogeneous structures. These findings indicate that continuous-time quantum walks, when combined with topology-independent decoupling strategies, provide a powerful paradigm for large-scale combinatorial optimization and complex network control, with potential applications spanning infrastructure resilience, epidemic containment, sensor network optimization, and biological systems analysis.

Scalable Quantum Walk-Based Heuristics for the Minimum Vertex Cover Problem

TL;DR

This work addresses the NP-hard Minimum Vertex Cover problem by introducing a scalable, resource-efficient heuristic based on continuous-time quantum walks. By encoding vertices with qubits and evolving under a graph-encoded normalized Laplacian Hamiltonian, the method uses short-time CTQW dynamics to identify influential vertices via transition probabilities, followed by a dynamic decoupling (freezing) step to remove selected vertices from further interference. The approach achieves superior approximation ratios across diverse graph ensembles (ER, BA, REG) and exhibits robustness to topology, outperforming classical heuristics and MILP baselines in many cases, while benefiting from an exponential qubit reduction. This topology-agnostic, quantum-inspired framework has potential implications for large-scale network optimization and related applications in resilience, containment, and infrastructure planning, with feasible pathways toward quantum or hybrid implementations.

Abstract

We propose a novel heuristic quantum algorithm for the Minimum Vertex Cover (MVC) problem based on continuous-time quantum walks (CTQWs). In this framework, the coherent propagation of a quantum walker over a graph encodes its structural properties into state amplitudes, enabling the identification of highly influential vertices through their transition probabilities. To enhance stability and solution quality, we introduce a dynamic decoupling (``freezing'') mechanism that isolates vertices already selected for the cover, preventing their interference in subsequent iterations of the algorithm. The method employs a compact binary encoding, requiring only qubits to represent a graph with vertices, resulting in an exponential reduction of quantum resources compared to conventional vertex-based encodings. We benchmark the proposed heuristic against exact solutions obtained via Mixed-Integer Linear Programming (MILP) and against established classical heuristics, including Simulated Annealing, FastVC, and the 2-Approximation algorithm, across Erdős--Rényi, Barabási--Albert and regular random graph ensembles. Our results demonstrate that the CTQW-based heuristic consistently achieves superior approximation ratios and exhibits remarkable robustness with respect to network topology, outperforming classical approaches in both heterogeneous and homogeneous structures. These findings indicate that continuous-time quantum walks, when combined with topology-independent decoupling strategies, provide a powerful paradigm for large-scale combinatorial optimization and complex network control, with potential applications spanning infrastructure resilience, epidemic containment, sensor network optimization, and biological systems analysis.

Paper Structure

This paper contains 10 sections, 10 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Hamiltonian and Graph Encoding
  2. Quantum Walk Dynamics and Transition Probabilities
  3. Modeling Vertex Removal with Dynamic Decoupling
  4. Iterative Heuristics
  5. Quantum heuristic formulation
  6. Classical simulation of the quantum walk
  7. Erdős–Rényi graphs.
  8. Barabási–Albert graphs.
  9. Regular graphs.
  10. Topological dependence.

Figures (3)

  • Figure 1: Iterative Quantum Walk Protocol for MVC. The figure illustrates the sequential algorithm for determining the MVC. (a) An initial graph is defined. (b) The graph is mapped onto quantum states using binary encoding, and the system is evolved via a Quantum Walk. (c-d) Transition probabilities between states are calculated, and the state with the highest probability is identified and saved. (e-g) Iterative Decoupling: The saved state is decoupled from the remaining system, which is then evolved again. The next state with the highest transition probability is selected and saved. (h-j) This decoupling and selection process continues iteratively, identifying the next most probable state. (k) Final Solution: The iteration stops when all remaining states are decoupled (i.e., no more couplings exist between them). The collection of all saved states represents the MVC for the initial graph.
  • Figure 2: Heatmap showing the average approximation ratio for the Quantum, FastVC, Simulated Annealing (SA), and 2-Approximation algorithms. The ratio is defined as the size of the algorithm's solution divided by the exact MVC size, which was computed using Mixed-Integer Linear Programming (MILP). Mean values were calculated over a set of graph instances for each topology. Ratios closer to $1.0$ indicate superior average performance and higher solution quality.
  • Figure 3: Average Performance and Statistical Dispersion of Approximation Ratios as a Function of Graph Size. The figure displays the average approximation ratio achieved by the Quantum (blue), FastVC (orange), Simulated Annealing (SA) (green), and 2-Approximation (red) algorithms for the MVC problem as a function of the number of nodes (N). The analysis is segmented across three graph topologies: Erdos-Renyi, Barabasi-Albert, and Regular. The data was generated from $10$ randomized graph instances per node count $(N)$. The central line represents the mean ratio, while the shaded region indicates the $\pm 1$ standard deviation $(\sigma)$, reflecting the statistical dispersion of the algorithms' performance due to the stochastic nature of the graph generation. The approximation ratio is defined relative to the exact MVC, obtained via the MILP solver. Ratios closer to $1.0$ (indicated by the dashed line) denote superior average performance.