Variable Parameter Analysis for Scheduling One Machine

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 $\nu$, $\nu<<n$. 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 $\mathsf{NP}$-hard single-machine scheduling problem to minimize maximum job lateness. The target variable parameter $\nu$ 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 $n-\nu$ non-emerging jobs is constructed in a low degree polynomial time. At phase 2 less than $\nu!$ permutations of the $\nu$ 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, $\nu/n$ 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 $n$s.