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

We study a sequential decision-making problem on a $n$-node graph $\mathcal{G}$ where each node has an unknown label from a finite set $\mathbf{\Sigma}$, drawn from a joint distribution $\mathcal{P}$ that is Markov with respect to $\mathcal{G}$.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 $\mathcal{G}$ is a forest.Our implementation runs in $\mathcal{O}(n^2 \cdot |\mathbf{\Sigma}|^2)$ time while using $\mathcal{O}(n \cdot |\mathbf{\Sigma}|^2)$ oracle calls to $\mathcal{P}$ and $\mathcal{O}(n^2 \cdot |\mathbf{\Sigma}|)$ 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.