This paper describes a robust approach for the single machine scheduling problem 1|r i |L max . The method is said to be robust since it characterizes a large set of optimal solutions allowing to switch from one solution to another, without any performance loss, in order to face potential disruptions which occur during the schedule execution. It is based on a dominance theorem that characterizes a set of dominant sequences, using the interval structure defined by the relative order of the release and the due dates of jobs. The performance of a set of dominant sequences can be determined in polynomial time by computing the most favorable and the most unfavorable sequences associated with each job, with regard to the lateness criterion. A branch and bound procedure is proposed which modifies the interval structure of the problem in order to tighten the dominant set of sequences so that only the optimal sequences are conserved.
This paper focuses on the characterization of a subset of optimal sequences for the famous two-machine flowshop problem. Based on the relative order of the job processing times, two particular interval structures are defined so that each job is associated with an interval. Then, using the Allen's algebra, the interval relationships are analysed and a sufficient optimality condition is established providing a characterisation of a large subset of optimal sequences. This set necessarily includes any Johnson's sequences together with numerous other optimal job sequences.
A 2-distance k-coloring of a graph is a proper vertex k-coloring where vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance 4-coloring for planar subcubic graphs with girth at least 21. We also show a construction of a planar subcubic graph of girth 11 that is not 2-distance 4-colorable.
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