Maximizing Concave Functions in Fixed Dimension

In the authors introduced a technique which enabled them to solve the parametric minimum cycle problem with a xed number of parameters in strongly polynomial time In the current paper we present this technique as a general tool In order to allow for an independent reading of this paper we repeat some of the de nitions and propositions given in Some proofs are not repeated however and instead we supply the interested reader with appropriate pointers Suppose Q R 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 a ne algo rithm 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 op erations required by A on any input i e point in Q is T We show that under these assumptions for any xed d the function g can be maximized in a number of operations polynomial in k and T We also present a general frame work for parametric extensions of problems where this technique can be used to obtain strongly polynomial algorithms Norton Plotkin and Tardos applied a similar scheme and presented additional applications