13 research outputs found
A polynomial oracle-time algorithm for convex integer minimization
In this paper we consider the solution of certain convex integer minimization
problems via greedy augmentation procedures. We show that a greedy augmentation
procedure that employs only directions from certain Graver bases needs only
polynomially many augmentation steps to solve the given problem. We extend
these results to convex -fold integer minimization problems and to convex
2-stage stochastic integer minimization problems. Finally, we present some
applications of convex -fold integer minimization problems for which our
approach provides polynomial time solution algorithms.Comment: 19 pages, 1 figur
On the complexity of nonlinear mixed-integer optimization
This is a survey on the computational complexity of nonlinear mixed-integer
optimization. It highlights a selection of important topics, ranging from
incomputability results that arise from number theory and logic, to recently
obtained fully polynomial time approximation schemes in fixed dimension, and to
strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear
Optimization, IMA Volumes, Springer-Verla
Graphs of Transportation Polytopes
This paper discusses properties of the graphs of 2-way and 3-way
transportation polytopes, in particular, their possible numbers of vertices and
their diameters. Our main results include a quadratic bound on the diameter of
axial 3-way transportation polytopes and a catalogue of non-degenerate
transportation polytopes of small sizes. The catalogue disproves five
conjectures about these polyhedra stated in the monograph by Yemelichev et al.
(1984). It also allowed us to discover some new results. For example, we prove
that the number of vertices of an transportation polytope is a
multiple of the greatest common divisor of and .Comment: 29 pages, 7 figures. Final version. Improvements to the exposition of
several lemmas and the upper bound in Theorem 1.1 is improved by a factor of
tw