School network reorganization under educational and spatial constraints using classical and quantum optimization

Technical Summary: School Network Reorganization under Educational and Spatial Constraints

Problem Definition
The paper addresses the strategic reorganization of school networks, a complex combinatorial optimization problem driven by demographic decline, fiscal constraints, and the need for territorial equity. Specifically, the problem involves determining which autonomous school institutions should retain their administrative status and which should be aggregated into larger "school hubs." This decision must balance conflicting objectives: optimizing administrative efficiency and resource allocation while preserving equitable access to education, respecting institutional regulations, and maintaining social cohesion in fragile, geographically isolated territories. The study focuses on the Italian context, where school dimensioning policies are governed by specific legislative frameworks regarding institutional compatibility, provincial boundaries, and enrollment thresholds.

Methodology
The authors propose a unified optimization framework based on Integer Linear Programming (ILP) that integrates six primary dimensions:

1. Administrative Geography: Aggregations are restricted to the same province; cross-municipal aggregations are penalized to favor local governance.
2. Institutional Compatibility: Schools can only aggregate if they belong to the same educational category (Comprehensive Institutes vs. Upper Secondary Institutes). Within Upper Secondary Institutes, curricular track mismatches are treated as soft constraints with associated penalties.
3. Territorial Fragility: Municipalities are classified by a criticality index (levels 0–4) based on income, education, labor, and governance. Aggregations in highly critical areas (levels 3 and 4) are heavily penalized to prevent further marginalization.
4. Dimensional Requirements: Eligibility for reorganization is determined by enrollment thresholds that vary inversely with territorial criticality (e.g., schools in the most fragile areas require fewer students to remain autonomous).
5. Spatial Accessibility: Aggregations are feasible only if the travel time between schools (computed on the road network) does not exceed category-specific thresholds.
6. Operational Feasibility: Constraints ensure that aggregated schools do not exceed capacity limits and that smaller schools are absorbed only by larger or equal-sized peers.

The paper presents two mathematical formulations:

• Variant I (Baseline ILP): A standard formulation where compatibility and accessibility constraints are enforced via explicit linear constraints.
• Variant II (Capacitated Facility Location Problem - CFLP): A more compact reformulation where compatibility requirements are embedded directly into the domain of the assignment variables, reducing the constraint structure.

To evaluate the model, the authors developed a synthetic benchmark generator that reproduces realistic regional school networks with hierarchical settlement patterns and varying demographic scenarios. Additionally, a real-world case study was conducted using the complete public school network of the Calabria region (Italy).

The study investigates both classical and quantum optimization approaches. Classical optimization utilizes the Gurobi Optimizer (exact solver). Quantum optimization employs a hybrid quantum-classical framework (D-Wave Leap) using a Constrained Quadratic Model (CQM) to solve the problem via Quantum Annealing (QA).

Key Contributions

• Novel ILP Formulation: The paper introduces a specific ILP model for school dimensioning that uniquely integrates institutional regulations, road-network travel times, territorial fragility, and curricular compatibility within a single framework.
• Synthetic Benchmark Generation: The authors developed a generator for synthetic instances, addressing the lack of public benchmark datasets for school dimensioning problems.
• Quantum Optimization Proof-of-Concept: The work reformulates the school dimensioning problem as a CQM and implements it on a hybrid quantum solver, assessing the feasibility of emerging quantum technologies for public-service network planning.
• Real-World Validation: The model is applied to the entire Calabria school network, demonstrating its applicability to complex, real-world operational scenarios involving 190 decision variables and diverse territorial constraints.

Results

• Classical Performance: The compact CFLP formulation (Variant II) demonstrated high computational efficiency. For synthetic instances ranging from 250 to 1,000 schools, the Gurobi solver found optimal solutions in negligible time (average runtime < 0.12 seconds for 1,000 schools). Sensitivity analysis revealed that geographical coherence (inter-municipal penalties) and enrollment thresholds were the dominant drivers of the optimization, while territorial protection and accessibility thresholds had minimal statistical influence on the optimal objective value in the tested synthetic scenarios.
• Quantum Performance: In the hybrid quantum-classical experiments (tested on the 500-school synthetic instance and the Calabria case study), the solver consistently reproduced the exact optimal solutions found by the classical solver with a 0% optimality gap and zero standard deviation across repeated runs. However, the quantum approach exhibited significantly higher execution times (approx. 11–13 seconds) compared to the classical solver (< 0.2 seconds), attributed to the overhead of the hybrid framework rather than the optimization difficulty itself.
• Calabria Case Study: The model successfully generated optimal aggregation plans for the Calabria region under 64 different policy configurations. The results showed structural robustness; variations in policy weights did not drastically alter the territorial organization or the number of educational hubs, suggesting the network's geography is the primary determinant of the solution structure.

Significance and Claims
The paper claims that the proposed framework serves as a robust decision-support tool for regional authorities, enabling evidence-based planning that balances administrative efficiency with territorial equity. The authors emphasize that the methodology effectively preserves the structural characteristics of the educational system while identifying optimal aggregation plans under diverse policy scenarios.

Regarding quantum computing, the paper maintains a modest stance. It does not claim computational superiority over classical methods. Instead, it positions the work as a validation of the proposed mathematical formulation's compatibility with emerging quantum technologies. The results demonstrate that the school dimensioning problem possesses the structural characteristics (binary variables, sparse constraints) suitable for hybrid quantum optimization. The authors conclude that while current quantum hardware is not yet faster than state-of-the-art classical solvers for this specific problem, the successful reproduction of optimal solutions confirms the viability of the approach as quantum hardware matures. Future research directions suggested include multi-period planning, demographic forecasting, and the application of next-generation quantum hardware to larger-scale instances.