We consider a generalization of the dining philosophers problem to arbitrary connection topologies. We focus on symmetric, fully distributed systems, and we address the problem of guaranteeing progress and lockout-freedom, even in presence of adversary schedulers, by using randomized algorithms. We show that the well-known algorithms of Lehmann and Rabin do not work in the generalized case, and we propose an alternative algorithm based on the idea of letting the philosophers assign a random priority to their adjacent forks.