Maximizing concave functions in fixed dimension (1993) [12 citations — 0 self]
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@INPROCEEDINGS{Cohen93maximizingconcave,author = {Edith Cohen and Nimrod Megiddo},
title = {Maximizing concave functions in fixed dimension},
booktitle = {in: Complexity in Numeric Computation},
year = {1993},
pages = {74--87},
publisher = {World Scientific Press}
}
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Abstract:
In [3, 5, 2] 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 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 de nitions 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 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 a ne 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 inQ) is T. We show that under these assumptions, for any xedd, the function g can be maximized in a number of operations polynomial in k and T.We also present a general framework for parametric extensions of problems where this technique can be used to obtain strongly polynomial algorithms. Norton, Plotkin, and Tardos [12] applied a similar scheme and presented additional applications. Keywords: Complexity, concave-cost network ow, capacitated, global optimization, local optimization. 1 See [4, 2] for an earlier version. 1 1.
