  We consider how to use the Bellman residual of the dynamic programming
operator to compute suboptimality bounds for solutions to stochastic shortest
path problems. Such bounds have been previously established only in the special
case that "all policies are proper," in which case the dynamic programming
operator is known to be a contraction, and have been shown to be easily
computable only in the more limited special case of discounting. Under the
condition that transition costs are positive, we show that suboptimality bounds
can be easily computed even when not all policies are proper. In the general
case when there are no restrictions on transition costs, the analysis is more
complex. But we present preliminary results that show such bounds are possible.
