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Convex Integer Maximization via Graver Bases

J. De Loera·R. Hemmecke·S. Onn·U. G. Rothblum·R. Weismantel
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Abstract

We present a new algebraic algorithmic scheme to solve {\em convex integer maximization} problems of the following form, where $c$ is a convex function on $R^d$ and $w_1x,...,w_dx$ are linear forms on $R^n$, $$\max \{c(w_1 x,...,w_d x): Ax=b, x\in N^n\} .$$ This method works for arbitrary input data $A,b,d,w_1,...,w_d,c$. Moreover, for fixed $d$ and several important classes of programs in {\em variable dimension}, we prove that our algorithm runs in {\em polynomial time}. As a consequence, we obtain polynomial time algorithms for various types of multi-way transportation problems, packing problems, and partitioning problems in variable dimension.

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J. De Loera, R. Hemmecke, S. Onn, U. G. Rothblum, R. Weismantel. 2006-09-01. Convex Integer Maximization via Graver Bases. https://arxiv.org/abs/math/0609019

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