On polynomial solvability of the high multiplicity total weighted tardiness problem

AbstractIn a recent paper Hochbaum developed a polynomial algorithm for solving a scheduling problem of minimizing the total weighted tardiness for a large number of unit length jobs which can be partitioned into few sets of jobs with identical due dates and penalty weights. The number of unit jobs in a set is called the multiplicity of that set. The problem was formulated in Hochbaum as an integer quadratic nonseparable transportation problem and solved, in polynomial time, independent of the size of the multiplicities and the due dates but depending on the penalty weights. In this paper we show how to solve the above problem in polynomial time which is independent of the sizes of the weights. The running time of the algorithm depends on the dimension of the problem and only the size of the maximal difference between two consecutive due dates. In the case where the due dates are large, but the size of the maximal difference between two consecutive due dates is polynomially bounded by the dimension of the problem, the algorithm runs in strongly polynomial time

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

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