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An FPTAS for optimizing a class of low-rank functions over a polytope

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Abstract

We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of non-linear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Pareto-optimal front of the linear functions which constitute the given low-rank function. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasi-concavity on the objective function. For the special case of quasi-concave function minimization, we give an alternative FPTAS, which always returns a solution which is an extreme point of the polytope. Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with non-linear objective functions, for example when the objective is a product of a fixed number of linear functions. We also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. We prove this by showing a similar hardness of approximation result for supermodular function minimization, a result that may be of independent interest.

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Authors and Affiliations

  1. Amazon.com, 333 Boren Avenue N, Seattle, WA, 98109, USA

    Shashi Mittal

  2. Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, E62-569, Cambridge, MA, 02139, USA

    Andreas S. Schulz

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  1. Shashi Mittal
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  2. Andreas S. Schulz
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Correspondence to Shashi Mittal.

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Mittal, S., Schulz, A.S. An FPTAS for optimizing a class of low-rank functions over a polytope. Math. Program. 141, 103–120 (2013). https://doi.org/10.1007/s10107-011-0511-x

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