Daniel Kowalczyk

Constant angle surfaces in product spaces

Abstract We classify all the surfaces in M 2 ( c 1 ) × M 2 ( c 2 ) for which the tangent space T p M 2 makes constant angles with T p ( M 2 ( c 1 ) × { p 2 } ) (or equivalently with T p ( { p 1 } × M 2 ( c 2 ) ) for every point p = ( p 1 , p 2 ) of M 2 . Here M 2 ( c 1 ) and M 2 ( c 2 ) are 2 -dimensional space forms, not both flat. As a corollary we give a classification of all the totally geodesic surfaces in M 2 ( c 1 ) × M 2 ( c 2 ) .

An exact algorithm for parallel machine scheduling with conflicts

We consider an extension of classic parallel machine scheduling where a set of jobs is scheduled on identical parallel machines and an undirected conflict graph is part of the input. Each node in the graph represents a job, and an edge implies that its two jobs are conflicting, meaning that they cannot be scheduled on the same machine. The goal is to find an assignment of the jobs to the machines such that the maximum completion time (makespan) is minimized. We present an exact algorithm based on branch and price that combines methods from bin packing, scheduling, and graph coloring, with appropriate modifications. The algorithm has a good computational performance even for parallel machine scheduling without conflicting jobs.

ON EXTRINSICALLY SYMMETRIC HYPERSURFACES OF ℍ n ×ℝ

Totally umbilical, semi-parallel and parallel hypersurfaces of ℍ n ×ℝ are completely classified. More examples arise than in the analogous study on the ambient space 𝕊 n ×ℝ.

The robust machine availability problem – bin packing under uncertainty

Abstract We define and solve the robust machine availability problem in a parallel machine environment, which aims to minimize the number of identical machines required while completing all the jobs before a given deadline. The deterministic version of this problem essentially coincides with the bin packing problem. Our formulation preserves a user-defined robustness level regarding possible deviations in the job durations. For better computational performance, a branch-and-price procedure is proposed based on a set covering reformulation. We use zero-suppressed binary decision diagrams for solving the pricing problem, which enable us to manage the difficulty entailed by the robustness considerations as well as by extra constraints imposed by branching decisions. Computational results are reported that show the effectiveness of a pricing solver with zero-suppressed binary decision diagrams compared with a mixed integer programming solver.

The robust machine availability problem

We define and solve the robust machine availability problem in a parallel machine environment, which aims to minimize the number of identical machines required while completing all the jobs before a given deadline. Our formulation preserves a user-defined robustness level regarding possible deviations in the job durations. For better computational performance, a branch-and-price procedure is proposed based on a set covering reformulation. We use zero-suppressed binary decision diagrams (ZDDs) for solving the pricing problem, which enable us to manage the difficulty entailed by the robustness considerations as well as by extra constraints imposed by branching decisions. Computational results are reported that show the effectiveness of a pricing solver with ZDDs compared with a MIP solver.

Improving column generation methods or sheduling problems using ZDD and stabilization

In this work we tackle the parallel machine scheduling problem with identical machines to minimize the sum of weighted completion times. We study the set covering formulation for this problem that was introduced by van den Akker et al. [1], and improve the performance of their branch-and-price algorithm by a number of techniques, including zero-suppressed binary decision diagrams (ZDD) and stabilization. These techniques are sufficiently generic to be promising also for other scheduling problems.

A Branch-and-Price Algorithm for Parallel Machine Scheduling Using ZDDs and Generic Branching

We study the parallel machine scheduling problem to minimize the sum of the weighted completion times of the jobs to be scheduled (problem Pm||ΣwjCj in the standard three-field notation). We use the set covering formulation that was introduced by van den Akker et al. (1999) for this problem, and we improve the computational performance of their branch-and-price (B&P) algorithm by a number of techniques, including a different generic branching scheme, zero-suppressed binary decision diagrams (ZDDs) to solve the pricing problem, dual-price smoothing as a stabilization method, and Farkas pricing to handle infeasibilities. We report computational results that show the effectiveness of the algorithmic enhancements, which depends on the characteristics of the instances. To the best of our knowledge, we are also the first to use ZDDs to solve the pricing problem in a B&P algorithm for a scheduling problem.