Reliable assignments of processors to tasks and factoring on matroids

In the simple assignment problem, there are n processors, m tasks, and a relation between the processors and tasks; this relation indicates the ability of the processor to perform the task. When the processors fail independently with known probabilities, two performance issues arise. First, with what probability can the operating processors all be kept busy? Second, with what probability can the operating processors perform the same number of tasks that all processors could? We formulate these questions on the underlying transversal matroid. We first prove that counting minimum cardinality circuits in this matroid is #P-complete, and hence that both questions are also #P-complete. Secondly, we devise a factoring algorithm with series and parallel reductions to compute exact solutions of the above problems. We then outline some efficient strategies for bounding the probabilities.