Branch Less, Cut More and Schedule Jobs with Release and Delivery Times on Uniform Machines

We consider the problem of scheduling <i>n</i> jobs with identical processing times and given release as well as delivery times on <i>m</i> uniform machines. The goal is to minimize the makespan, i.e., the maximum full completion time of any job. This problem is well-known to have an open complexity status even if the number of jobs is fixed. We present a polynomial-time algorithm for the problem which is based on the earlier introduced algorithmic framework blesscmore (“branch less and cut more”). We extend the analysis of the so-called behavior alternatives developed earlier for the version of the problem with identical parallel machines and show how the earlier used technique for identical machines can be extended to the uniform machine environment if a special condition on the job parameters is imposed. The time complexity of the proposed algorithm is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mi>γ</mi><msup><mi>m</mi><mn>2</mn></msup><mi>n</mi><mo form="prefix">log</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> can be either <i>n</i> or the maximum job delivery time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><mo movablelimits="true" form="prefix">max</mo></msub></semantics></math></inline-formula>. This complexity can even be reduced further by using a smaller number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>κ</mi><mo><</mo><mi>n</mi></mrow></semantics></math></inline-formula> in the estimation describing the number of jobs of particular types. However, this number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula> becomes only known when the algorithm has terminated.