The Complexity of Reachability in Randomized Sabotage Games

We analyze a model of fault-tolerant systems in a probabilistic setting. The model has been introduced under the name of “sabotage games”. A reachability problem over graphs is considered, where a “Runner” starts from a vertex u and seeks to reach some vertex in a target set F while, after each move, the adversary “Blocker” deletes one edge. Extending work by Löding and Rohde (who showed PSpacecompleteness of this reachability problem), we consider the randomized case (a “game against nature”) in which the deleted edges are chosen at random, each existing edge with the same probability. In this much weaker model, we show that, for any probability p and ε > 0, the following problem is again PSpace-complete: Given a game graph with u and F and a probability p′ in the interval [p− ε, p+ ε], is there a strategy for Runner to reach F with probability ≥ p′? Our result extends the PSpace-completeness of Papadimitriou’s “dynamic graph reliability”; there, the probabilities of edge failures may depend both on the edge and on the current position of Runner.