Extending partial suborders and implication classes

We consider the following problem called transitive ordering with precedence constraints (TOP): Given a partial order P=(V,&;) and an (undirected) graph G=(V,E) such that all relations in P are represented by edges in G. Is there a transitive orientation D=(V,A) of G, such that P is contained in D

More-Dimensional Packing with Order Constraints

We present a first systematic study on more-dimensional packing problems with order constraints. Problems of this type occur naturally in applications such as logistics or computer architecture. They can be interpreted as more-dimensional generalizations of scheduling problems. Using graph-theoretic structures to describe feasible solutions, we develop a novel exact branch-and-bound algorithm. This extends previous work by Fekete and Schepers; a key tool is a new order-theoretic characterization of feasible extensions of a partial order to a given complementarity graph that is tailor-made for use in a branch-and-bound environment. The usefulness of our approach is validated by computational results

Symmetry for Periodic Railway Timetables

Periodic timetabling for railway networks is usually modeled by the Periodic Event Scheduling Problem (PESP). This model permits to express many requirements that practitioners impose on periodic railway timetables. We discuss a requirement practitioners are asking for, but which, so far, has not been the topic of mathematical studies: the concept of symmetry

Algorithms for Manufacturing Paperclips and Sheet Metal Structures

We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a "carpenter's ruler") can be straightened, a problem that was open for several years and has only recently been solved in the affirmative

Scheduling Parallel Jobs to Minimize Makespan

We consider the NP-hard problem of scheduling parallel jobs with release dates on identical parallel machines to minimize the makespan. A parallel job requires simultaneously a pre-specified, job-dependent number of machines when being processed. Our main result is the following. The makespan of a (non-preemptive) schedule constructed by any listscheduling algorithm is within a factor of 2 of the optimal preemptive makespan. This gives the best known approximation algorithms for both the preemptive and the non-preemptive variant of the problem, improving upon previously known performance guarantees of 3. We also show that no listscheduling algorithm can achieve a better performance guarantee than 2 for the non-preemptive problem, no matter which priority list is chosen. Since listscheduling also works in the online setting in which jobs arrive over time and the length of a job becomes only known when it completes, the main result yields a deterministic online algorithm with competitive ratio 2 as well. In addition, we consider a different online model in which jobs arrive one by one and need to be scheduled before the next job becomes known. In this context, no listscheduling algorithm has a constant competitive ratio. We present the first online algorithm for scheduling parallel jobs with a constant competitive ratio. We also prove a new information-theoretic lower bound of 2:25 for the competitive ratio of any deterministic online algorithm for this model