Convex Integer Maximization via Graver Bases

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 $w_{1}x,...,w_{d}x$ 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.