Abstract maximum flow with partial switching

Abstract network is the generalization of classical network topology which is associated with the set of elements and linearly ordered subset of elements, known as paths. Each path of an abstract network must satisfy the switching property, that is, if two paths cross at a common element then there must be a path that is a subset of the first path up to the crossing element and subset of the second path after the crossing element. Each element is equipped with integral movement capacity which transships the flow from an element to its adjacent element through a path. In this paper, we introduce the concept of the partial switching property, where the presence of residual paths, in combination with fully switched paths, eliminates the crossing effect and increases the flow value. We introduce both the maximum static and dynamic flow problems with partial path switching and present their corresponding flow models. To solve these problems, we propose polynomial time algorithms. Additionally, we extend these results to the contraflow problem and present a polynomial-time solution strategy. Finally, we extend the abstract flow and contraflow problems with partial switching to continuous time settings and solve them using natural transformation technique.