Capacity factors in a point-to-point network

In this study, we investigate some properties of capacity factors (CFs) that may affect the link failure problem in network coding. Roughly speaking, a CF is an edge set where its deletion will reduce the maximum flow, whereas deleting any proper subset of this CF will not. More generally, a k-CF is a minimal (not minimum) edge set, where its removal will decrease the maximum network flow by at least k. First, under a point-to-point acyclic scenario, we characterize all the edges contained in some CFs and we propose an efficient classification algorithm. We show that all of the edges on some s–t path in a point-to-point acyclic network belong to some 2-CF, and we also present some other properties of CFs. Some computational hardness results related to CFs are obtained. We prove that determining the existence of a CF in a cyclic network with a size greater than a given number is an NP-complete problem and that the time complexity of computing its capacity rank is lower bounded by that of solving its maximal flow. In addition, we propose an analogous definition of CFs on vertices as a special case that captures edge CFs.