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Convex integer maximization via Graver bases. (English) Zbl 1284.05026
Summary: We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where \(c\) is a convex function on \(\mathbb R^d\) and \(w_{1}x,\dots ,w_dx\) are linear forms on \(\mathbb R^n\), \[ \max\{c(w_1x,\dots , w_dx): Ax = b, x \in \mathbb N^n\}. \] This method works for arbitrary input data \(A,b,d,w_{1},\dots ,w_{d},c\). 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.

05A17 Combinatorial aspects of partitions of integers
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
Full Text: DOI arXiv
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