36 Computational Convexity

The subject of Computational Convexity draws its methods from discrete mathematics and convex geometry, and many of its problems from operations research, computer science, data analysis, physics, material science, and other applied areas. In essence, it is the study of the computational and algorithmic aspects of high-dimensional convex sets (especially polytopes), with a view to applying the knowledge gained to convex bodies that arise in other mathematical disciplines or in the mathematical modeling of problems from outside mathematics. The name Computational Convexity is of more recent origin, having first appeared in print in 1989. However, results that retrospectively belong to this area go back a long way. In particular, many of the basic ideas of Linear Programming have an essentially geometric character and fit very well into the conception of Computational Convexity. The same is true of the subject of Polyhedral Combinatorics and of the Algorithmic Theory of Polytopes and Convex Bodies. The emphasis in Computational Convexity is on problems whose underlying structure is the convex geometry of normed vector spaces of finite but generally not restricted dimension, rather than of fixed dimension. This leads to closer connections with the optimization problems that arise in a wide variety of disciplines. Further, in the study of Computational Convexity, the underlying model of computation is mainly the binary (Turing machine) model that is common in studies of computational complexity. This requirement is imposed by prospective applications, particularly in mathematical programming. For the study of algorithmic aspects of convex bodies that are not polytopes, the binary model is often augmented by additional devices called “oracles.” Some cases of interest involve other models of computation, but the present discussion focuses on aspects of computational convexity for which binary models seem most natural. Many of the results stated in this chapter are qualitative, in the sense that they classify certain problems as being solvable in polynomial time, or show that certain problems are NP-hard or harder. Typically, the tasks remain to find optimal exact algorithms for the problems that are polynomially solvable, and to find useful approximation algorithms or heuristics for those that are NP-hard. In many cases, the known algorithms, even when they run in polynomial time, appear to be far from optimal from the viewpoint of practical application. Hence, the qualitative complexity results should in many cases be regarded as a guide to future efforts but not as final words on the problems with which they deal. Some of the important areas of computational convexity, such as linear and convex programming, packing and covering, and geometric reconstructions, are covered in other chapters of this Handbook. Hence, after some remarks on presentations of polytopes in Section 36.1, the present discussion concentrates on the following areas that are not covered elsewhere in the Handbook: 36.2, Algorithmic Theory