Budgeted sensor placement for source localization on trees

This paper considers the problem of source localization on graphs. We assume that a diffusion process on a graph is partially observed, i.e., that only the time at which some of the vertices are reached by the diffusion is known, and we want to identify the vertex that initiated the diffusion. In particular, we address the problem of choosing a fixed number of sensor vertices based on whose diffusion time one can optimally estimate the source. Building on the definition of Double Resolving Set of a graph, we introduce a notion of vertex resolvability that turns out to be the key to study the optimality of a sensor set. For the case of trees we give polynomial time algorithms for both allocating a fixed budget k of sensors such that they maximize the probability of correct detection of the source, and for identifying the k sensors that minimize the expected distance between the real source and the estimated one.