The Assumption-based Truth Maintenance System (ATMS) is a general and powerful problem-solving tool in AI. Unfortunately, its generality usually entails a high computational cost. In this paper, we study how a general notion of cost function can be incorporated into the design of an algorithm for focusing the ATMS, called BF-ATMS. The BF-ATMS algorithm explores a search space of size polynomial in the number of assumptions, even for problems which are proven to have exponential size labels. Experimental results indicate significant speedups over the standard ATMS for such problems. In addition to its improved efficiency, the BF-ATMS algorithm retains the multiple-context capability of an ATMS, and the important properties of consistency, minimality, soundness, as well as the property of bounded completeness. The usefulness of the new algorithm is demonstrated by its application to the task of consistency-based diagnosis, where dramatic efficiency improvements, with respect to the standard solution technique, are obtained.