Variable Parameter Analysis for Scheduling One Machine
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In contrast to the fixed parameter analysis (FPA), in the variable parameter analysis (VPA) the value of the target problem parameter is not fixed, it rather depends on the structure of a given problem instance and tends to have a favorable asymptotic behavior when the size of the input increases. While applying the VPA to an intractable optimization problem with n objects, the exponential-time dependence in enumeration of the feasible solution set is attributed solely to the variable parameter ν, ν<<n. As opposed to the FPA, the VPA does not imply any restriction on some problem parameters, it rather takes an advantage of a favorable nature of the problem, which permits to reduce the cost of enumeration of the solution space. Our main technical contribution is a variable parameter algorithm for a strongly 𝖭𝖯-hard single-machine scheduling problem to minimize maximum job lateness. The target variable parameter ν is the number of jobs with some specific characteristics, the “emerging” ones. The solution process is separated in two phases. At phase 1 a partial solution including n-ν non-emerging jobs is constructed in a low degree polynomial time. At phase 2 less than ν! permutations of the ν emerging jobs are considered. Each of them are incorporated into the partial schedule of phase 1. Doe to the results of an earlier conducted experimental study, ν/n varied from 1/4 for small problem instances to 1/10 for the largest tested problem instances, so that that the ratio becomes closer to 0 for large ns.
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