On Learning Optimal Corners in Orthogonal Partially Observable Cooperative Guard Art Galleries

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.

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.

Constrained Multi-agent reinforcement learning (CMARL) faces two intertwined challenges: the joint action space grows exponentially with the number of agents, and additional requirements couple agents in ways that reward structure alone does not capture. We introduce Coordination Graphs for Constrained Multi-Agent Reinforcement Learning (CG-CMARL), a framework that addresses both challenges by combining coordination graphs with Lagrangian duality. The system decomposes the joint problem into pairwise regions, each served by a set of shared Q-functions, one for the primary objective and one for each of the constraints, so that the number of learned models is independent of the number of agents. At execution time, Max-Sum message passing coordinates actions across the factor graph, while a Lagrangian multiplier controls the objective--constraint tradeoff, allowing a single trained model to trace a Pareto front without retraining. We provide convergence guarantees under mild conditions, together with a compositional error bound that decomposes into separate interpretable sources, each traceable to a specific design choice and independently controllable. Experiments on cooperative navigation tasks (where teams of up to 10 agents must coordinate to reach target positions while satisfying pairwise constraints) show that our method produces Pareto fronts dominating established baselines trained at fixed reward-shaping ratios, while scaling to team sizes where centralized approaches become intractable.