Efficient methods for determining the lower and upper bounds on the probabilities of source-to-terminal communication in a multistage interconnection network are developed. A novel lower bounding strategy (shifting) and a novel upper bounding strategy (renormalization) are presented; both can be computed in polynomial time. These strategies can be combined with existing methods based on coherence, and on consecutive cuts, to obtain an improvement on previously known efficiently computable bounds. A second efficient upper bound (averaging) is developed. An empirical evaluation of the bounds is discussed. Finally, the value of these bounding strategies in assessing the reliability of interconnection networks is examined. >