Graph planning with expected finite horizon

Fixed-horizon planning considers a weighted graph and asks to construct path that maximizes the sum of weights for given time horizon T. However, in many scenarios, is not fixed, but stopping chosen according some distribution such expected If stopping-time known, then ensure robustness, by an adversary as worst-case scenario. A stationary plan every vertex always chooses same outgoing edge. For fixed or distribution, plans are sufficient optimality. Quite surprisingly we show when with T, sufficient. While computing optimal NP-complete, under adversarial can be achieved polynomial time.