Some Algorithmic Problems in Polytope Theory

Convex polyhedra, i.e., the intersections of finitely many closed affine halfspaces inRd, are important objects in various areas of mathematics and other disciplines. In particular, the compact ones among them (polytopes), which equivalently can be defined as the convex hulls of finitely many points in Rd, have been studied since ancient times (e.g., the platonic solids). Polytopes appear as building blocks of more complicated structures, e.g., in (combinatorial) topology, numerical mathematics, or computer aided design. Even in physics polytopes are relevant (e.g., in crystallography or string theory). Probably the most important reason for the tremendous growth of interest in the theory of convex polyhedra in the second half of the 20 century was the fact that linear programming (i.e., optimizing a linear function over the solutions of a system of linear inequalities) became a widespread tool to solve practical problems in industry (and military). Dantzig’s Simplex Algorithm, developed in the late 40’s, showed that geometric and combinatorial knowledge of polyhedra (as the domains of linear programming problems) is quite helpful for finding and analyzing solution procedures for linear programming problems. Since the interest in the theory of convex polyhedra to a large extent comes from algorithmic problems, it is not surprising that many algorithmic questions on polyhedra arose in the past. But also inherently, convex polyhedra (in particular: polytopes) give rise to algorithmic questions, because they can be treated as finite objects by definition. This makes it possible to investigate (the smaller ones among) them by computer programs (like the polymake-system written by Gawrilow and Joswig, see [26] and [27,28]). Once chosen to exploit this possibility, one immediately finds oneself confronted with many algorithmic challenges. This paper contains descriptions of 35 algorithmic problems about polyhedra. The goal is to collect for each problem the current knowledge about its