We investigate a family of inference problems on Markov models, where many sample paths are drawn from a Markov chain and partial information is revealed to an observer who attempts to reconstruct the sample paths. We present algorithms and hardness results for several variants of this problem which arise by revealing different information to the observer and imposing different requirements for the reconstruction of sample paths. Our algorithms are analogous to the classical Viterbi algorithm for Hidden Markov Models, which finds single most probable sample path given a sequence of observations. Our work is motivated by an important application in ecology: inferring bird migration paths from a large database of observations.