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).


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.

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Last update: August 16, 2004.