Abstract

In [3, 5, 2] the authors introduced a technique which enabled them to solve the parametric minimum cycle problem with a fixed number of parameters in strongly polynomial time. In the current paper 1 we present this technique as a general tool. In order to allow for an independent reading of this paper, we repeat some of the definitions and propositions given in [3, 5, 2]. Some proofs are not repeated, however, and instead we supply the interested reader with appropriate pointers. Suppose Q ae R d is a convex set given as an intersection of k halfspaces, and let g : Q ! R be a concave function that is computable by a piecewise affine algorithm (i.e., roughly, an algorithm that performs only multiplications by scalars, additions, and comparisons of intermediate values which depend on the input). Assume that such an algorithm A is given and the maximal number of operations required by A on any input (i.e., point in Q) is T . We show that under these assumptions, for any fixed d, the ...

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