Abstract
The problem models the scenario where there is set of jobs to be processed on a single machine, and each job i can only be scheduled for processing in exactly one time interval from a group \(G_i\) of allowed intervals. The objective is to determine if there is a set of \(S \subseteq [\gamma ]\) of \(k\ (k \in \mathbb {N})\) jobs which can be scheduled in non-overlapping time intervals.
This work describes a deterministic algorithm for the problem that runs in time \({\text {O}}({(5.18)}^k n^d)\), where \(n = |\bigcup _{i \in [\gamma ]} G_i|\) and \(d \in \mathbb {N}\) is a constant. For \(k \ge d \log {n}\), this is significantly faster than the best previously-known deterministic algorithm, which runs in time \({\text {O}}({(12.8)}^k \gamma n)\). We obtain our speedup using efficient constructions of representative families, which can be used to solve the problem by a dynamic programming approach.
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Biswas, A., Raman, V., Saurabh, S. (2019). Solving Group Interval Scheduling Efficiently. In: Colbourn, C., Grossi, R., Pisanti, N. (eds) Combinatorial Algorithms. IWOCA 2019. Lecture Notes in Computer Science(), vol 11638. Springer, Cham. https://doi.org/10.1007/978-3-030-25005-8_9
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