Peter Kleinschmidt

The d-Step Conjecture and Its Relatives

The d-step conjecture arose from an attempt to understand the computational complexity of edge-following algorithms for linear programming, such as the simplex algorithm. It can be stated in terms of diameters of graphs of convex polytopes, in terms of the existence of nonrevisiting paths in such graphs, in terms of an exchange process for simplicial bases of a vector space, and in terms of matrix pivot operations. First formulated by W. M. Hirsch in 1957, the conjecture remains unsettled, though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures. If the conjecture is false, as we believe to be the case, then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture. This report summarizes what is known about the d-step conjecture and its relatives. A considerable amount of new material is included, but it does not seem to come close to settling the conjecture. Of special interest is the first example of a polytope that is not vertex-decomposable, showing that a certain natural approach to the conjecture will not work. Also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture.

A constrained matching problem

We show that certain manpower scheduling problems can be modeled as the following constrained matching problem. Given an undirected graphG = (V,E) with edge weights and a digraphD = (V,A). AMaster/Slave-matching (MS-matching) ofG with respect toD is a matching ofG such that for each arc (u, v) εA for which the nodeu is matched, the nodev is matched, too. TheMS-Matching Problem is the problem of finding a maximum-weight MS-matching. Letk(D) be the maximum size of a (weakly) connected component ofD. We prove that MS-matching is an NP-hard problem even ifG is bipartite andk(D) ≤ 3. Moreover, we show that in the relevant special case wherek(D) ≤ 2, the MS-Matching Problem can be transformed to the ordinary Matching Problem.

Reconstructing a Simple Polytope from its Graph

Let P be a simple polytope with dimension d and G(P) its edge graph. It has been shown in [BlM87] and [Kal88] that G(P) already determines the complete face-lattice of P. However, the constructive approach used in [Kal88] requires the computation of all orderings in vert(P)1 which is computationally prohibitive for polytopes of even very small sizes. In this paper we propose an algorithm which is still exponential but does work with reasonable computing time for non-trivial simple polytopes.

On the existence of certain smooth toric varieties

We prove that the combinatorial types of those cone systems which correspond to complete smooth toric varieties are more restricted than for complete toric varieties: the toric varieties corresponding to essentially all types of cyclic polytopes possess singularities. This yields a negative answer to a problem stated by G. Ewald. Some consequences and problems concerning mathematical programming and the rational cohomology of smooth toric varieties are discussed.

Reflections on the architecture of a decision support system for personnel assignment scheduling in production cell technology

Abstract Within the range of new technology applications ‘flexible manufacturing systems’ are expected to give a productivity edge in market competition. Their function is based on a decentralized manufacturing organisation in form of semi-autonomous workgroups. These workgroups, consisting of a small number of workers, form so-called production cells. In this paper we want to present considerations on the architecture of a decision support system for personnel assignment scheduling within production cell organisation. Beside this, social, organisational, and qualificational aspects and requirements for the use of such systems are discussed.

A Polyhedral Approach for a Constrained Matching Problem

Abstract. Certain manpower scheduling problems can be solved as weighted mat\-ching problems with additional constraints. A complete linear description of a polytope whose vertices correspond to such matchings is provided and how this can be used for solving more complex problems with a cutting plane procedure is described.

Computing an Upper Bound for the Longest Edge in an Optimal TSP-Solution

A solution of the traveling salesman problem (TSP) with n nodes consists of n edges which form a shortest tour. In our approach we compute an upper bound u for the longest edge which could be in an optimal solution. This means that every edge longer than this bound cannot be in an optimal solution. The quantity u can be computed in polynomial time. We have applied our approach to different problems of the TSPLIB (library of sample instances for the TSP). Our bound does not necessarily improve the fastest TSP-algorithms. However, the reduction of the number of edges might be useful for certain instances.