Planner-Admissible Graph-PDE Value Extensions for Sparse Goal-Conditioned Planning

Sparse goal-conditioned planning with few cost-to-go labels can be viewed as a graph-PDE Dirichlet extension problem: extend sparse labels on a goal-dependent boundary to unlabelled graph vertices so that greedy rollouts reach the goal. We study which graph value extensions are planner-admissible under the operational argmin-Q planner. Our main result is a local action-gap certificate: if the surrogate value error along the rollout stays below half the true action gap, then the greedy rollout reaches the goal. Absolutely Minimal Lipschitz Extension (AMLE), the p=infinity endpoint of the graph p-Laplacian family, instantiates this certificate through a comparison-principle fill-distance bo...

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