## 1 Introduction  
In the stable marriage problem of [PERSON] and [PERSON] [1], there are \(n\) men and \(n\) women, and each person ranks the members of the opposite gender arbitrarily according to an individual strict preference order. A _marriage scheme_ in this model is a set of marriages between different men and women. Such a scheme is _unstable_ if there exists a _blocking pair_, that is, a man \(m\) and a woman \(w\), such that \(m\) is either unmarried or \(m\) prefers \(w\) to his wife, and at the same time, \(w\) is either unmarried or prefers \(m\) to her partner. A marriage scheme is _stable_ if it is not unstable, that is, not blocked by any pair. It is a natural problem to find a stable marriage scheme if it exists at all. Nowadays, it is already folklore that for any preference rankings of the \(n\) men and \(n\) women, a stable marriage scheme does exist. This theorem was proved first by [PERSON] and [PERSON] in [1]. They constructed a special stable marriage scheme with the help of a finite procedure, the so-called deferred acceptance algorithm. It also turned out that for the existence of a stable scheme, it is not necessary that the number of men is the same as the number of women or that for each person, all members of the opposite group are acceptable: the deferred acceptance algorithm is so robust, that it works properly under these more general conditions. Several interesting properties of the structure of stable marriage schemes are known. [PERSON] [2] attributes to [PERSON] the observation that stable marriages have a lattice structure: if each man picks the better assignment out of two stable marriage schemes, then another stable marriage scheme is created in which each woman receives the worst partner out of the two. There are further known extensions of the stable marriage problem. [PERSON] and [PERSON] proved in [3] that if each edge of the underlying bipartite graph has a nonnegative capacity and each vertex has a nonnegative quota, then the accordingly modified deferred acceptance algorithm always finds a so-called stable allocation. An allocation is an assignment of nonnegative values to the edges that do not exceed the corresponding capacities, such that the total allocation of no vertex exceeds its quota. In other words, an allocation is a marriage scheme where a bipartite marriage can be formed with an \"intensity\" different from zero and one, and each participant has an individual upper bound on his/her total \"marriage intensity\". An allocation is stable if any unsaturated edge, \(e\), has a saturated end vertex, \(v\), such that each edge, \(e^{\prime}\), incident with \(v\) has an assigned value of zero whenever \(v\) prefers \(e\) to \(e^{\prime}\). That is, if the intensity of a marriage is not maximum, then one of the spouses has maximum total marriage intensity, and none of his/her marriages is worse than the particular marriage. Beyond proving the existence of stable assignments, [PERSON] and [PERSON] used flow-type arguments to speed up the deferred acceptance algorithm in [3]. Later, [PERSON] and [PERSON] came up with an even faster network flow based algorithm for the same problem [4]. It is fairly well-known that the bipartite matching problem can be formulated in the more general network flow model, and the alternating path algorithm for maximum bipartite matchings is a special case of the augmenting path algorithm of [PERSON] and [PERSON] for maximum flows. However, it seems that the question of whether there exists a flow generalization of the stable marriage theorem has not been addressed so far. This very problem is the focus of our present work. It turned out that our model is closely related to so-called \"supply chains\" well known in the economics literature. Prior to our work, [PERSON] published a related result in [5]. The main difference between [PERSON]'s and our model is that [PERSON]'s one is acyclic and discrete, whereas ours allows cycles and has a fractional formulation, as well.