Minimum Parametric Flow – A Partitioning Approach

The present paper proposes a partitioning type approach for the parametric minimum flow problem which is based on the classical decreasing directed paths method. On each of its iterations, the algorithm finds a decreasing directed path from source node to sink node in a range of parametric residual networks which are consecutively defined for subintervals of the parameter values and, by decreasing the flow along the corresponding paths in the original parametric network, splits the interval of the parameter values in subintervals generated by the breakpoints of the piecewise linear parametric residual capacity function of the decreasing directed path. Further on, the algorithm reiterates for every generated subinterval in increasing order of the parameter values.


INTRODUCTION
The problem of the parametric maximum flow with zero lower bounds and linear capacity functions has constantly been investigated and several algorithms exist (e.g. Hamacher and Foulds [1], Ruhe [2,3], Gallo et al. [4] or Zhang et al. [5,6]) to solve different instances the problem. Although it has its own applications, the parametric minimum flow problem was

Method Article
addressed in literature considerably less often than the parametric maximum flow problem. Actually, the parametric minimum flow problem (or, in general any parametric flow problem) represents a kind of generalisation of the classical nonparametric problem to the case where the lower bounds (or correspondingly the upper bounds) of some arcs depend of a certain parameter. Consequently, the problem consists in solving the nonparametric minimum flow problem for all the parameter values within a certain interval. If all the lower bound functions linearly depend of the parameter, the minimum parametric flow value function will result in a continuous piecewise linear function of the same parameter. The partitioning type approach, which is presented in this paper, proposes an original algorithm for computing the minimum flow in networks with linear upper bound functions. As Bichot and Siarry [7] showed, the parametric flow problem "is of genuine practical and theoretical interest since graph partitioning applications are described on a wide variety of subjects as: data distribution in parallelcomputing, VLSI circuit design, image processing, computer vision, route planning, air traffic control, mobile networks, social networks, etc" [7].
The structure of this article is the following: Section 2 gathers some basic terminology elements regarding the network flow problem. The terminology and definitions in this section are taken from [8]. Section 3 reminds some necessary definitions regarding the parametric minimum flow problem. Section 4 suggests a possible application of the parametric minimum flow problem. Section 5 describes the proposed algorithm which finds a parametric minimum flow. Section 6 gives an example of a parametric network with linear lower bounds functions in order to show the evolution of the proposed algorithm. Finally, Section 7 presents some conclusions and a generalization of the problem.

TERMINOLOGY AND PRELIMINARIES
In the previous equation (1)

PARAMETRIC MINIMUM FLOW
The parametric minimum flow problem can be regarded as a generalisation of the nonparametric problem where the lower bounds of some arcs depend of a nonnegative, real parameter λ : In expression (2), by . The same restriction also holds for the lower bound of every arc . Definitions 1 to 7, which follow in this section, are adapted from reference [8] while theorem 1 is taken from reference [9].
In the previous definition (4) (5) and is defined as: ) , ( j i

APPLICATIONS
Considering that the problem of the parametric minimum flow represents an extension of the classical problem of the minimum flow, the applications of the parametric minimum flows cover all those instances of the classical flow problem where network characteristics are depending linearly of a parameter. This subsection briefly presents the problem of scheduling some works on different machines. The problem has many practical applications if the machines are considered to be workers, oil ship, freight trucks, wagons, airplanes or even processors etc.
Let X be the set of works that must be made by a set Y of machines. Each work X x i ∈ is performed by a machine is also added to the network.
The optimal scheduling of the work can be found by solving a minimum flow problem in the parametric network described above.

PARTITIONING ALGORITHM FOR FINDING THE PARAMETRIC MINIMUM FLOW
Every iteration of the proposed algorithm achieves an improvement of the flow within a subinterval of the parameter values. This subinterval is constantly updated so that in its inside the residual capacities of all network arcs show no breakpoints. A range of parametric residual networks that take into account the previously stated requirement are successively defined by the algorithm. By doing so, the difficulty to compare or to subtract two piecewise linear functions can be avoided.
As soon as the parametric residual network k G ′ contains no directed path from the source node to the sink node, the algorithm reiterates over the next subinterval, until the whole interval of the parameter values is covered. Due to the fact that build the parametric residual capacity ) (P r′ of the directed path P; (9) update the value of i.e.
For this phase, see the algorithms presented in the papers of Ahuja, Magnanti and Orlin [10] or Ciurea and Ciupală [11]. Further on, with 0 = k the algorithm builds the parametric residual

EXAMPLE
For the network presented in Fig. 1(a)