Parameterized complexity of machine scheduling: 15 open problems

Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc

Throughput Rate Optimization in High Multiplicity Sequencing Problems

Mixed model assembly systems assemble products (parts) of differenttypes in certain prespecified quantities. A minimal part set is a smallestpossible set of product type quantities, to be called the multiplicities,in which the numbers of assembled products of the various types are inthe desired ratios. It is common practice to repeatedly assemble minimalpart sets, and in such a way that the products of each of the minimalpart sets are assembled in the same sequence. Little is known howeverregarding the resulting throughput rate, in particular in comparison to thethroughput rates attainable by other input strategies. This paper investigatesthroughput and balancing issues in repetitive manufacturing environments.It considers sequencing problems that occur in this setting andhow the repetition strategy influences throughput. We model the problemsas a generalization of the traveling salesman problem and derive ourresults in this general setting. Our analysis uses well known concepts fromscheduling theory and combinatorial optimization.Economics ;

Polynomiality for Bin Packing with a Constant Number of Item Types

We consider the bin packing problem with d different item sizes s_i and item multiplicities a_i, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem. In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d >= 3. In fact, for constant d our algorithm solves the following problem in polynomial time: given two d-dimensional polytopes P and Q, find the smallest number of integer points in P whose sum lies in Q. Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant

On strongly polynomial algorithms for some classes of quadratic programming problems

In this paper we survey some results concerning polynomial and/or strongly polynomial solvability of some classes of quadratic programming problems. The discussion on polynomial solvability of continuous convex quadratic programming is followed by a couple of models for quadratic integer programming which, due to their special structure, allow polynomial (or even strongly polynomial) solvability. The theoretical merit of those results stems from the fact that a running time (i.e. the number of elementary arithmetic operations) of a strongly polynomial algorithm is independent of the input size of the problem