  The paper addresses two variants of the stochastic shortest path problem
('optimize the accumulated weight until reaching a goal state') in Markov
decision processes (MDPs) with integer weights. The first variant optimizes
partial expected accumulated weights, where paths not leading to a goal state
are assigned weight 0, while the second variant considers conditional expected
accumulated weights, where the probability mass is redistributed to paths
reaching the goal. Both variants constitute useful approaches to the analysis
of systems without guarantees on the occurrence of an event of interest
(reaching a goal state), but have only been studied in structures with
non-negative weights. Our main results are as follows. There are
polynomial-time algorithms to check the finiteness of the supremum of the
partial or conditional expectations in MDPs with arbitrary integer weights. If
finite, then optimal weight-based deterministic schedulers exist. In contrast
to the setting of non-negative weights, optimal schedulers can need infinite
memory and their value can be irrational. However, the optimal value can be
approximated up to an absolute error of $\epsilon$ in time exponential in the
size of the MDP and polynomial in $\log(1/\epsilon)$.
