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We present new filtering algorithms for Disjunctive and Cumulative constraints, each of which improves the complexity of the state-of-theart algorithms by a factor of log n. We show how to perform Time-Tabling and Detectable Precedences in linear time on the Disjunctive constraint. Furthermore, we present a linear-time Overload Checking for the Disjunctive and Cumulative constraints. Finally, we show how the rule of Not-first/Not-last can be enforced in quadratic time for the Cumulative constraint. These algorithms rely on the union find data structure, from which we take advantage to introduce a new data structure that we call it time line. This data structure provides constant time operations that were previously implemented in logarithmic time by the Θ-tree data structure. Experiments show that these new algorithms are competitive even for a small number of tasks and outperform existing algorithms as the number of tasks increases. We also show that the time line can be used to solve specific scheduling problems.

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The Cumulative constraint greatly contributes to the success of constraint programming at solving scheduling problems. The SoftCumulative, a version of the Cumulative where overloading the resource incurs a penalty is, however, less studied. We introduce a checker and a filtering algorithm for the SoftCumulative, which are inspired by the powerful energetic reasoning rule for the Cumulative. Both algorithms can be used with classic linear penalty function, but also with a quadratic penalty function, where the penalty of overloading the resource increases quadratically with the amount of the overload. We show that these algorithms are more general than existing algorithms and vastly outperform a decomposition of the SoftCumulative in practice.