## Bernd Fruhwirth, Rainer E. Burkard, and Günter Rote:

# Approximation of convex curves with application to the bicriterial minimum
cost flow problem

*European Journal of Operational Research*
**42** (1989), 326–338,
(MR #91e:90107).
### Abstract

An approximation of an explicitly or implicitly given convex curve in the
plane is given by two piecewise linear “outer” and “inner” curves. To
compute these, three rules for choosing the supporting points are proposed and
it is shown for two of them that the projective distance between inner and
outer approximation decreases quadratically with the number of supporting
points. These two rules are variations of the Sandwich algorithm. The third
rule takes a more local approach. It is a compromise between the adaptive
global strategy of the Sandwich algorithm and the usual local left-to-right
parametric approach for computing the efficient point curve.
This method is applied to approximate the efficient point curve of the
bicriteria minimum cost flow problem, which is a piecewise linear convex curve
that may have an exponential number of breakpoints in the worst case.

It turns out that the left-to-right rule is better from a computational
viewpoint, since is tries to avoid large changes in the parameters when
solving successive minimum cost flow problems.

Last update: August 16, 2004.