On Bounds for the Monotone-Structure Reliability

At present, two types of attainable bounds on the reliability of a general monotone structure are known. Packing bounds are trivial. Untying bounds were obtained by Polesskii in 1997. Their special case, the classical Esary–Proschan bounds, are known since 1963. In the present paper, we introduce the third type of attainable bounds on the reliability of a general monotone structure. We call them difference-untying bounds. They are generalization and improvement of the Oxley–Welsh bounds on the reliability of a homogeneous monotone structure obtained in 1979. An example demonstrating the high quality of difference-untying bounds is given. As a consequence, we obtain new bounds on the number of members of an arbitrary simplicial complex.