arXiv:2211.02107 · v2

Variable Parameter Analysis for Scheduling One Machine

首次提交 2022-11-04 03:22

版本更新 2026-11-03 07:38

雷达观测 2026-11-04 23:40

摘要

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 $\mathsf{NP}$-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 $n$s.

作者备注：arXiv admin note: substantial text overlap with arXiv:2103.09900