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A cumulative not-first/not-last filtering algorithm in \(O(n^2 \log(n))\). (English) Zbl 1282.90070
Summary: In cumulative and disjunctive constraint-based scheduling, the resource constraint is enforced by several filtering rules. Among these rules, we have (extended) edge-finding and not-first/not-last rules. The not-first/not-last rule detects tasks that cannot run first/last relatively to a set of tasks and prunes their time bounds. In this paper, it is presented a sound \(O(n^2 \log(n))\) algorithm for the cumulative not-first/not-last rule where \(n\) is the number of tasks. This algorithm reaches the same fix point as previous not-first/not-last algorithms, although it may take additional iterations to do so. The worst case complexity of this new algorithm for the maximal adjustment is the same as our previous complete \(O(n^2|H| \log n)\) not-first/not-last algorithm [the authors, “A Complete Filtering Algorithm for Cumulative Not-First/Not-Last rule in \(O(n^2|H| \log n)\)”, in: Proceeding of CSCLP 2010, Berlin, Germany, 31–42 (2010)] where \(|H|\) is the maximum between the number of distinct earliest completion and latest start times of tasks. But, experimental results on benchmarks from the Project Scheduling Problem Library (PSPLib) and the Baptiste and Le Pape data set (BL) suggest that the new not-first/not-last algorithm has a substantially reduced runtime. Furthermore, the results demonstrate that in practice the new algorithm rarely requires more propagations than previous not-first/not-last algorithms.

MSC:
90B35 Deterministic scheduling theory in operations research
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