Partitioning Algorithm for the Parametric Maximum Flow

Abstract

The article presents an approach to the maximum flow problem in parametric networks with linear capacity functions of a single parameter, based on the concept of shortest conditional augmenting directed path. In order to avoid working with piecewise linear functions, our approach uses a series of parametric residual networks defined for successive subintervals of the parameter values where the parametric residual capacities of all arcs remain linear functions. Besides working with linear instead piecewise linear functions, another main advantage of our approach is that every directed path in such a parametric residual network is also a conditional augmenting directed path for the subinterval for which the parametric residual network was defined. The complexity of the partitioning algorithm is where  2 O Kn m  K is the number of partitioning points of the parameter values interval, n and m being the number of nodes, respectively the number of arcs in the network.

Parametric flows in static networks

“…Definitions included in the following sections are taken from/adapted after, references [18] and [17] while the proof for theorems can be found in reference [19].…”

“…The "Partitioning algorithm" for the parametric maximum flow [18] / minimum flow [17] problem represents an original approach of the directed paths type algorithms. From a feasible flow, established in the first stage of the algorithm, the partitioning algorithm finds, on every iteration of its second stage, an augmentation/decreasing directed path from the source node to the sink node in a parametric residual network defined only for a subinterval of the parameter values, it improves the flow along the corresponding paths in the original parametric network and splits the interval of the parameter values into subintervals which are generated by the breakpoints of the piecewise linear parametric residual capacity function of the augmentation/decreasing directed path.…”

A Preflow Approach to Maximum Flows in Parametric Networks Applied to Assessing the Legal Information

“…Section 2 presents the parametric maximum flow problem and the basic parametric network flow terminology and results which are used in the rest of the paper. Most of the definitions in this section are straightforward modifications of those in [4] and [6], adapted for the maximum flow problem in parametric networks with zero lower bounds. More specialized terminology and further details on the notions, definitions and main results within this section can be found in [1], [4] and [6].…”

“…Most of the definitions in this section are straightforward modifications of those in [4] and [6], adapted for the maximum flow problem in parametric networks with zero lower bounds. More specialized terminology and further details on the notions, definitions and main results within this section can be found in [1], [4] and [6]. In Section 3 we present our algorithm for solving the parametric maximum flow problem, called "Highest label partitioning push (HLPP) algorithm".…”

“…Partitioning algorithms for the parametric maximum flow problem can be developed starting from any of the three classic approaches for the maximum flow problem: flow augmenting directed paths algorithms, preflow algorithms or min-max algorithm. The approach based on flow augmenting directed paths [4], makes use of the concept of shortest conditional augmenting directed path. In order to avoid working with piecewise linear functions, the approach uses a series of parametric residual networks defined for successive subintervals of the parameter values where the parametric residual capacities of all arcs remain linear functions.…”