Genetic Programming with Behaviour-based Niching for Learning Guided Local Search in Vehicle Routing Problems

Genetic Programming Guided Local Search (GPGLS) uses genetic programming to evolve utility functions for guided local search in large-scale vehicle routing problems (LSVRPs). Evaluating every GP individual on every training instance at every generation is expensive, so GPGLS is usually trained on small instance batches. Existing curriculum-based GPGLS orders these batches mainly by instance size. Adaptive Curriculum Learning GPGLS (ACL-GPGLS) improves training efficiency by adapting when the search moves between fixed curriculum stages, but the instance difficulty order remains predefined. We propose DCL-GPGLS, which estimates the difficulty of each training instance from the current population's solution quality and updates the estimates during evolution. Each generation then receives a batch near a scheduled difficulty level, with a correction that limits repeated selection of the same instances. Experiments on a fixed training-test split of the CVRPLIB X set show that DCL-GPGLS achieves the best observed average rank and mean test cost among six training policies. It obtains the lowest mean cost on 36 of 65 unseen test instances and is significantly better than the static feedback-derived curriculum, matched in total evaluator calls, on 6 instances, with no significant difference on the remaining 59.

Although Vehicle Routing Problems (VRP) are essential to many real-world systems, they remain computationally intractable at scale due to their combinatorial complexity. Traditional heuristics rely on handcrafted rules for local improvements and occasional \textit{jumps} to escape local minima, but often struggle to generalize across diverse instances. We introduce \textbf{COAgents}, a cooperative multi-agent framework that models the search process as a graph: nodes represent solutions, and edges correspond to either local refinements or large perturbations for diversification (i.e., jumps). A \textit{Partial Search Graph} (PSG) is dynamically constructed during search, enabling COAgents to train a Node Selection Agent and a Move Selection Agent to guide intensification, and a Jump Agent to trigger well-timed explorations of new regions. Unlike end-to-end learning approaches, COAgents cleanly separates problem-agnostic search control from compact domain-specific encoding, facilitating adaptability across tasks. Extensive experiments on the CVRP and VRPTW benchmarks show that COAgents remains competitive with several learn-to-search baselines on CVRP and sets a new state of the art among learning-based methods on the more challenging VRPTW instances, reducing the gap to the best-known solutions by 14% at $N\!=\!100$ and 44% at $N\!=\!50$ relative to the strongest neural solver (POMO), and by 21% and 40% respectively relative to ALNS. Code is available at https://github.com/mahdims/COAgents.

Graph coarsening reduces the large Quadratic Unconstrained Binary Optimization (QUBO) formulations arising when vehicle-routing problems are solved by quantum annealing. Nearby customers with compatible time windows are merged into super-nodes, the reduced problem is solved, and the solution is expanded to the original graph. For the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW), existing coarsening heuristics require family-specific tuning and remain unreliable on random instances. We address these limitations on the Solomon benchmark using simulated annealing and a D-Wave Advantage2 processor. We first introduce adaptive penalty calibration. Uniform penalty scaling has little effect, whereas controlling the internal coefficient range substantially improves raw samples. Removing non-binding constraints, normalising binding ones, and scaling the remaining penalties reduces mean raw constraint violations from 33.0 to 0.06 at the same solver budget (p=3.7e-11, n=56). A variable-count-preserving control attributes this gain to conditioning rather than problem size. Second, we replace the hand-tuned merge score with a graph neural network (GNN) using one configuration across all families. At N=10, it achieves 100% feasibility across all Solomon families, including R-type (100% vs. 80% for the tuned heuristic). Across N=10,...,100, feasibility is 83% vs. 69%, with the GNN better or tied on 85/90 instance-size pairs. At N=80,100, the difference is significant (p=0.002; 25/25 pairs), while the QUBO remains approximately 5-6 times smaller. Finally, hardware experiments reproduce the conditioning effect at fixed logical variable count: feasible samples increase from 0.02% to 39% across 13 instances. Classical repair with local search remains a reference bound for end-to-end solution cost.