A hybrid quantum-classical neural network for learning to route

This work studies hybrid quantum-classical neural networks for learning routing heuristics. Specifically, this paper asks whether small quantum neural networks can replace parameter-heavy modules inside a competitive attention-based routing model while maintaining solution quality. For the capacitated vehicle routing problem, encoder feed-forward replacement emerges as the most promising design: it reduces the number of model parameters by 56.6% while keeping the hybrid model close to the classical neural baseline at small and medium instance sizes, although the gap grows for larger instances. This work also compares to classical routing algorithms, which remain highly competitive and often superior on the fixed Euclidean test sets. Our results therefore do not indicate quantum advantage or solver dominance, but identify encoder feed-forward replacement as a viable hybrid-module compression strategy for neural combinatorial optimization.

Sampling Luck Masquerades as Allocation Gain: Auditing Test-Time Budget Allocation for Neural Combinatorial Optimization

Neural combinatorial optimization (NCO) solvers report the best of many sampled solutions per instance, and the sample count is, by convention, identical for every instance. Whether a non-uniform allocation of a fixed total budget would buy anything has not been measured. We measure it, and we audit the measurement itself. First, on in-distribution workloads the allocation headroom is not detectable. Across three pretrained solvers (POMO, AM, SymNCO) on uniform TSP-100, an oracle allocation computed and evaluated on the same stored samples reports a 2.2-2.6% gain with intervals excluding zero; measured out of sample the same gain is indistinguishable from zero (0.457, 0.015, -0.512 percent). Following the customary in-sample procedure, all three solvers would have supported a published 2%-level gain that does not exist. We calibrate this bias against an instance-wise null in which the true gain is zero by construction; over the ranges we test it does not shrink with more samples or more instances. Second, the same correction that removes the phantom gains preserves a real one. Under distribution shift (a workload mixing uniform and clustered instances), a pre-registered confirmatory experiment finds that allocation guided by held-out sample statistics improves best-of-k by 11.5% (AM, primary endpoint; 95% CI [7.4, 19.7]) and 12.0% (SymNCO, replication) at equal evaluation budget, with the signal-acquisition cost not charged; a pre-registered negative control (POMO, an order of magnitude more robust to shift) shows -0.3% [-0.7, 0.24]. The gain exceeds a frozen distribution-label baseline by 4.2 points [1.9, 7.7]. An exploratory policy charging a 20-sample probe against the same budget retains 3.4% (AM) and 4.6% (SymNCO). We give a correction procedure and a reporting checklist, and release all data, code, and the pre-registration record.

DualCert: A Solver for the Traveling Salesman Problem with Constraint-Coupled Learning

Large traveling salesman problem (TSP) instances require a solver to allocate limited computation while preserving the validity of its outputs. Existing neural--operations-research (OR) hybrids predict guidance without requiring learned transitions to satisfy constraints discovered during search. DualCert introduces \emph{constraint-coupled learning}, in which current degree equations and dynamically separated subtour-elimination constraints (SECs) define each learned transition. At each refinement, the degree equations and selected, strictly satisfied SEC equations, with positive slacks, define an iterate-dependent primal-slack Karush--Kuhn--Tucker (KKT) manifold. Repaired dual variables and violated SEC rows define a local cost field. An exact constrained mirror-descent step maps each finite state to a positive state on the same manifold. Where selected rows and deterministic ties remain fixed, implicit differentiation maps parameter perturbations into the manifold tangent space and reuses the forward constraint operator for the local-cost-field derivative. The terminal edge state allocates computation across Held--Karp ascent, candidate-graph edge tests, and tour construction under a fixed budget. Deterministic verification recomputes original costs and accepts only verified candidate-graph lower bounds and edge decisions. On 1,000 held-out TSP1000 instances, DualCert attains a mean tour-cost gap of \(0.0573\%\) from Lin--Kernighan--Helsgaun version 3 (LKH-3) reference tours in \(9.55\) batch-amortized seconds per instance. It returns a verified candidate-graph lower bound for every instance and achieves \(81.46\%\) edge-decision coverage. The mean gap is \(67.1\%\) smaller than the reported NeuroLKH mean gap. Thus, optimization constraints govern learning, while deterministic verification preserves output validity.

AutoPref: Automatic Discovery of Task-Specific Preference Objectives for Neural Combinatorial Optimization

Combinatorial optimization problems (COPs) underpin many real-world decisions, but their exponentially large search spaces make high-quality solutions costly to obtain. Neural combinatorial optimization (NCO) learns fast construction policies, typically with reinforcement learning (RL), while preference-based NCO improves sample efficiency by learning from relative solution quality. However, existing preference objectives combine two distinct design choices in manually specified, one-size-fits-all formulations: what learning signal to extract from each solution pair and how to weight each pair relative to the sampled set. We present AutoPref, the first LLM-guided framework for automated preference-objective discovery in NCO. AutoPref factorizes the objective into a pairwise loss program, which defines the learning signal, and a set-aware weighting program, which determines each pair's relative contribution. Their composition forms a unified programmatic objective space containing existing preference objectives as special cases. To make its search tractable, we introduce a staged conditional search strategy with behavioral gates that filter inadmissible programs before short-horizon training and evaluation. Across TSP, CVRP, FFSP, and JSSP, AutoPref consistently outperforms strong hand-designed baselines across problem scales, demonstrating the benefits and scalability of automated objective discovery for NCO.

Learning to Place Guards by Reinforcement: A Geo-Free Neural Policy for the Vertex-Guard Art Gallery Problem

Neural combinatorial optimization (NCO) has shown that policies trained by reinforcement can construct strong solutions to NP-hard problems directly from raw instances. What such a policy actually learns, as opposed to what its decoder expresses, remains much less clear. We study this distinction on the vertex-guard Art Gallery Problem, the NP-hard task of choosing polygon vertices from which to observe an entire region. A pointer-network policy is trained from a coverage-aware reward over its own rollouts under the constraint we call geo-free inference: at test time it sees only vertex coordinates, with no visibility computation and no geometric oracle. The policy places guards economically but leaves a tail of under-covered polygons that widens far beyond the training range. To locate the cause, we freeze the trained encoder and read its embeddings with a small single-shot classifier, still geo-free at inference. The classifier closes most of the feasibility gap, in and out of distribution and at up to roughly five times the training range, cutting under-covered polygons by about an order of magnitude at an explicitly reported cost in guard count. We read this as evidence that the reinforcement-trained representation already encodes the geometry required for feasibility, and that residual failures reflect decoder calibration rather than missing knowledge. Probing a frozen encoder thus offers a practical way to ask what a neural combinatorial solver has internalized.

Interpreting Neural Combinatorial Optimization via Evolving Programmatic Bottlenecks

Neural Combinatorial Optimization (NCO) achieves strong performance, yet its black-box nature remains a key roadblock to deployment and scientific diagnosis. Standard interpretability tools, such as Concept Bottleneck Models (CBMs), are ill-equipped for NCO, whose decisions are dynamic, state-dependent, and lack proper concept vocabulary definition. To close this gap, we introduce Evolving Programmatic Bottlenecks (EPB), to our knowledge, the first framework for interpreting NCO policies by distilling black-box NCO models into human-readable program portfolios. EPB employs an LLM to autonomously evolve a bank of programs, where each program's per-step action distribution serves as the bottleneck. EPB works through an iterative framework: Block I fixes program bank capacity and introduces a hybrid textual-numerical gradient descent scheme that couples numerical gradients for student router updates and textual gradients for LLM-based program revision; Block II dynamically adapts bank capacity via fault-targeted expansion and redundancy pruning. Extensive experiments demonstrate EPB's effectiveness and broad applicability, where the distilled program portfolios largely match original performance. EPB also reveals that NCO behavior shifts across optimization stages and can be approximated as a composition of classic heuristic variants. Our work advances interpretable NCO and establishes EPB as a promising tool for interpreting sequential decision-making models.

N(CO)$^2$: Neural Combinatorial Optimization with Chance Constraints to Solve Stochastic Orienteering

Neural combinatorial optimization (NCO) offers a promising alternative to traditional heuristic-based methods for solving complex graph optimization problems by proposing to learn heuristics through data. This class of problems frequently arises in automation, as it can be used to model a variety of applications. While NCO has been extensively studied for deterministic combinatorial optimization problems, there are only a few works that aim to solve stochastic combinatorial optimization problems. In this work, we present N(CO)$^2$: Neural Combinatorial Optimization with Chance cOnstraints to solve the Stochastic Orienteering Problem (SOP) without the use of hand-crafted heuristics. By integrating a reinforcement learning (RL) framework, the model optimizes path selection under uncertainty, effectively balancing exploration and exploitation. Empirical results demonstrate that our method generalizes well across diverse SOP instances, achieving competitive performance compared to the state-of-the-art mixed-integer linear program (MILP) for the task. The proposed approach reduces human effort in heuristic design while enabling adaptive and efficient decision-making in uncertain environments.

Baseline-Free Policy Optimization for Neural Combinatorial Optimization

Neural combinatorial optimization (NCO) trains autoregressive policies to solve routing problems. The standard training algorithm, REINFORCE with a rollout baseline, requires maintaining and periodically updating a frozen copy of the policy for variance reduction. This baseline introduces a structural vulnerability: on harder instances, a poor baseline produces noisy gradient estimates that can destabilize training. We evaluate Group Relative Policy Optimization (GRPO), an algorithm from large language model alignment that eliminates the baseline entirely by normalizing advantages within groups of sampled trajectories. In a controlled comparison of five RL algorithms on TSP and CVRP benchmarks within the RL4CO framework, we find that: (i) GRPO avoids the training collapse observed with REINFORCE on TSP-100, where performance degrades from cost 9.8 to 52.1 immediately after the warmup phase and does not recover under extended training; (ii) at matched gradient updates, GRPO achieves solution quality within 2% of POMO, a strong AM-based multi-start baseline, while requiring no external baseline; and (iii) P3O, a pairwise preference algorithm also from the alignment literature, is competitive on TSP but shows higher variability on CVRP. These results identify GRPO as a promising baseline-free alternative for NCO, particularly in settings where baseline-dependent training becomes fragile.