Adaptive Frontier Exploration on Graphs with Applications to Network-Based Disease Testing

We study a sequential decision-making problem on a n-node graph G where each node has an unknown label from a finite set Omega, drawn from a joint distribution P that is Markov with respect to G. At each step, selecting a node reveals its label and yields a label-dependent reward. The goal is to adaptively choose nodes to maximize expected accumulated discounted rewards. We impose a frontier exploration constraint, where actions are limited to neighbors of previously selected nodes, reflecting practical constraints in settings such as contact tracing and robotic exploration. We design a Gittins index-based policy that applies to general graphs and is provably optimal when G is a forest. Our implementation runs in O(n2 · |Omega|2) time while using O(n · |Omega|2) oracle calls to P and O(n2 · |Omega|) space. Experiments on synthetic and real-world graphs show that our method consistently outperforms natural baselines, including in non-tree, budget-limited, and undiscounted settings. For example, in HIV testing simulations on real-world sexual interaction networks, our policy detects nearly all positive cases with only half the population tested, substantially outperforming other baselines.