The Vertex Set of a 0/1-Polytope is Strongly P-Enumerable

In this paper, we discuss the computational complexity of the following enumeration problem: Given a rational convex polyhedron P defined by a system of linear inequalities, output each vertex of P . It is still an open question whether there exists an algorithm for listing all vertices in running time polynomial in the input size and the output size. Informally speaking, a linear running time in the output size leads to the notion of P-enumerability introduced by Valiant [10]. The concept of strong P-enumerability additionally requires an output independent space complexity of the respective algorithm. We give such an algorithm for polytopes all of whose vertices are among the vertices of a polytope combinatorially equivalent to the hypercube. As a very important special case, this class of polytopes contains all 0/1-polytopes. Our implementation based on the commercial LP solver CPLEX 1 is superior to general vertex enumeration algorithms. We give an example how simplifications of..

Combinatorial Optimization Methods for Optimal Lines in Real-World Railway Systems (Extended Abstract)

1 Introduction In this paper we describe a problem which occurs in a railway system with periodic timetables. Nearly every urban public transportation system (tramway, bus) and a growing number of railway companies (e.g. Nederlandse Spoorwegen) use periodic timetables. In a railway system with periodic timetable a junction or line connecting two stations runs several times, in a fixed time interval (e.g. one hour), across the network. This number is called the frequency of the line. The problem considered in this paper consists of choosing some lines with their frequencies to serve passenger demand and to optimize a given objective. Several different objective functions are proposed. On one hand you may try to minimize operational costs for a fixed service [3], on the other hand you may wish to maximize service quality for fixed operational costs. One way to improve the service is to minimize the total travel time of all passengers. At this stage of planning there is no timetable, hen..

Yet Another Note on "An Efficient Zero-One Formulation of the Multilevel Lot-Sizing Problem"

In "An Efficient Zero-One Formulation of the Multilevel Lot-Sizing Problem" [8] MCKNEW,SAY- DAM,andCOLEMAN claim the polynomial solvability of this particular production planning problem. Both, the proof given by MCKNEW et al. and a statement of its incorrectness by RAJAGOPALAN [10] contain errors, and the question remains whether the approach leads to a polyomial time algorithm. We show by means of an example that solving the linear program is not sufficient

A Fast Algorithm For Near Cost Optimal Line Plans

We consider the design of line plans in public transport at a minimal total cost. Both, linear and nonlinear integer programming are adequate and intuitive modeling approaches for this problem. We present a heuristic variable fixing procedure which builds on problem knowledge from both techniques. We derive and compare lower bounds from different linearizations in order to assess the quality of our solutions. The involved integer linear programs are strengthened by means of problem specific valid inequalities. Computational results with practical data from the Dutch Railways indicate that our algorithm gives excellent solutions within minutes of computation time