Strongly polynomial algorithm for generalized flow maximization

A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of steps, an arc can be identified that must be tight in every dual optimal solution, and thus can be contracted. The full version is available on arXiv:1307.6809. The generalized flow model is a classical extension of network flows. Besides the capacity constraints, for every arc e there is a gain factor γe > 0, such that flow amount gets multiplied by γe while traversing the arc e. We study the flow maximization problem, where the objective is to send the maximum amount of flow to a sink node t. The model was already formulated by Kantorovich [17], as one of the first examples of linear programming; it has several applications in operations research [2, Chapter 15]. Gain factors can be used to model physical changes such as leakage or theft. Other common applications use the nodes to represent different types of entities, e.g. different currencies, and the gain factors correspond to the exchange rates. The existence of a strongly polynomial algorithm for linear programming is a major open question from a theoretical perspective. This refers to an algorithm with the number of arithmetic operations polynomially bounded in the number of variables and constraints, and the size of the numbers during the computations polynomially bounded in the input size. The landmark result by Tardos [27] gives an algorithm with the running time dependent only on the size of numbers in the constraint matrix, but independent from the right-hand side and the objective vector. This gives strongly polynomial algorithms for several combinatorial problems such as minimum cost flows (see also Tardos [26]) and multicommodity flows. Instead of the sizes of numbers, one might impose restrictions on the structure of the constraint matrix. Hence a natural question arises whether there exists a strongly polynomial algorithm for linear programs (LPs) with at most two nonzero entries per column (that can be arbitrary numbers). This question is still open; as shown by Hochbaum [15], all such LPs can be polynomially transformed to instances of the minimum cost generalized flow problem. (Note also that every LP can be polynomially transformed to an equivalent one with at most three nonzero entries per column.) Generalized flow maximization is an important special case of minimum cost generalized flows; it is probably the simplest natural class of LPs where no strongly polynomial algorithm has been known. The existence of such an algorithm has been a well-studied and longstanding open problem