Parametric Minimum Cost Flows¶
There exists a variety of solutions to the minimum cost flow problem. paminco provides solutions to a slighly different problem.
1. Minimum cost flow functions¶
The goal is to find a minimum cost flow function that maps a demand factor
where show definitions:
For notational simplicity, we only discuss linear demand functions
1.1 Alpha-beta-approximate minimum cost flow functions¶
It may not always be (computationally) feasible to compute an exact minimum cost flow function and the function has to be approximated.
Consider an interval
where
2. Parametric electrial flows¶
Instead of computing a minimum cost flow
where
2.1 Linear marginal cost¶
Assume that the marginal cost for all edges are linear and of the following form:
where show definitions:
At a glance
Linear marginal cost result in a linear minimum cost flow function.
Example: linear resistances show / hide
2.2 Piecewise linear marginal costs¶
The assumption of linear marginal cost restricts the applicability of the algorithm. We now discuss problems with piecewise linear marginal costs:
The piecewise linear marginal edge cost subdivide the potential space
In a region
Furhter, for all regions, there exist numbers
Hence, the main challenge lies in finding the correct regions
Idea behind the Electrial Flow Algorithm
Piecewise linear marginal cost result in a subdivision of the potential space
2.3 Interpolation of cost functions¶
The EFA algorithm is restricted to piecewise linear marginal cost functions
See also
MCAInterpolationRule
: Criteria for proper approximation of cost functions.
References
- WK21
Klimm M, Warode P (2021) “Parametric Computation of Minimum Cost Flows with Piecewise Quadratic Costs.” Mathematics of Operations Research. Available at https://www3.math.tu-berlin.de/disco/research/publications/pdf/KlimmWarode2021.pdf
- Roc84
Rockafellar, RT (1984) Network Flows and Monotropic Optimization (John Wiley and Sons, Hoboken, NJ)