Convex integer maximization via Graver bases

We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R and w1x, . . . , wdx are linear forms on R, max {c(w1x, . . . , wdx) : Ax = b, x ∈ N} . This method works for arbitrary input data A, b, d, w1, . . . , wd, c. Moreover, for fixed d and several important classes of programs in variable dimension, we prove that our algorithm runs in 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. keywords: Graver basis, Gröbner basis, Graver complexity, contingency table, transportation polytope, transportation problem, integer programming, discrete optimization, packing, cutting stock, partitioning, clustering, polyhedral combinatorics, convex optimization, computational complexity. AMS Subject Classification: 05A, 15A, 51M, 52A, 52B, 52C, 62H, 68Q, 68R, 68U, 68W, 90B, 90C