(Generalized) Convexity and Discrete Optimization
This short survey exhibits some of the important roles (generalized) convexity plays in integer programming. In particular integral polyhedra are discussed, the idea of polyhedral combinatorics is outlined and the use of convexity concepts in algorithmic design is shown. Moreover, combinatorial optimization problems arising from convex configurations in the plane are discussed.
KeywordsIntegral polyhedra polyhedral combinatorics integer programming convexity combinatorial optimization
Unable to display preview. Download preview PDF.
- Bank, B. and Mandel, R. (1988), (Mixed-) Integer solutions of quasiconvex polynomial inequalities. In: Advances in Mathematical Optimization, J. Guddat et al. (eds), Akademie Verlag, Berlin, pp. 20–34.Google Scholar
- Burkard, R.E. and Rudolf, R. (1993), Computational investigations on 3-dimensional axial assignment problems, Belgian J. of Operations Research, Vol. 32, 85–98.Google Scholar
- Christopher, G., Farach, M., and Trick, M. (1996), The structure of circular decomposable metrics, in: Algorithms — ESA’ 96, Lecture Notes in Comp. Sci., Vol. 1136, Springer, Berlin, pp.486–500.Google Scholar
- Hoffman, A. and Kruskal, J.B. (1956), Integral boundary points of convex polyhedra, in: Linear Inequalities and Related Studies, H. Kuhn and A. Tucker (eds.), Princeton University Press, Princeton, pp. 223–246.Google Scholar
- Karp, R.M. (1972), Reducibility among combinatorial problems, in: R.E. Miller and J.W. Thatcher (eds.), Complexity of Computer Computations, Plenum Press, New York, pp.85–103.Google Scholar
- Minkowski, H. (1896), Geometrie der Zahlen, Teubner, Leipzig.Google Scholar
- Preparata, F.P. and Shamos, M.I. (1988), Computational Geometry: An Introduction, Springer, Berlin.Google Scholar