Cost-based abduction attempts to find the best explanation for a set of facts by finding a minimal cost proof for the facts. The costs are computed by summing the costs of the assumptions necessary for the proof plus the cost of the rules. We examine existing methods for constructing explanations (proofs), as a minimization problem on a DAG (directed acyclic graph). We then define a probabilistic semantics for the costs, and prove the equivalence of the cost minimization problem to the Bayesian network MAP (maximum a posteriori probability) solution of the system. A simple best-first algorithm for finding least-cost proofs is presented, and possible improvements are suggested. The semantics of cost-based abduction for complete models are then generalized to handle negation. This, in turn, allows us to apply the best-first search algorithm as a novel way of computing MAP assignments to belief networks that can enumerate assignments in order of decreasing probability. An important point is that improvement results for the best-first search algorithm carry over to the computation of MAPs.