Parameterized Complexity of Partial Scheduling
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We study a natural variant of scheduling that we call partial scheduling: In this variant an instance of a scheduling problem along with an integer k is given and one seeks an optimal schedule where not all, but only k jobs have to be processed. We study the Fixed Parameter Tractability of partial scheduling problems parameterized by k for all variants of scheduling problems that minimize the makespan and involve unit/arbitrary processing times, identical/unrelated parallel machines, release/due dates, and precedence constraints. That is, we investigate whether algorithms with runtimes of the type O^*(f(k)) exist, where the O^*(·) notation omits factors polynomial in the input size. We obtain a trichotomy by categorizing each variant to be either in P, NP-complete and Fixed Parameter Tractable by k, or W[1]-hard by k. As one of our main technical contributions, we give an O^*(8^k) time algorithm to solve instances of k-scheduling problems minimizing the makespan with unit job lengths, precedence constraints and release dates.
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