Graph-PDE Value Extensions for Sparse Goal-Conditioned Planning

A recent preprint on arXiv (2605.19185) examines sparse goal-conditioned planning framed as a graph-PDE Dirichlet extension issue. In this context, sparse cost-to-go labels on a goal-specific boundary are propagated to unlabelled graph vertices, enabling greedy rollouts to achieve the goal. The authors propose a local action-gap certificate: if the surrogate value error during the rollout remains under half of the actual action gap, the greedy rollout will succeed. The Absolutely Minimal Lipschitz Extension (AMLE), which represents the p=infinity endpoint of the graph p-Laplacian series, validates this certificate via a fill-distance bound based on a comparison principle. Experiments with 120 configurations derived from AntMaze layouts indicate that while harmonic extension yields a 0.584 aggregate success rate, AMLE methods are likely to enhance planning reliability.